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<h1 class="libtitle">Auto<span class="subtitle">More Automation</span></h1>


<div class="code">

<span class="id" title="keyword">Set</span> <span class="id" title="var">Warnings</span> "-notation-overridden,-parsing,-deprecated-hint-without-locality".<br/>
<span class="id" title="keyword">From</span> <span class="id" title="var">Coq</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="http://coq.inria.fr/library//Coq.micromega.Lia.html#"><span class="id" title="library">Lia</span></a>.<br/>
<span class="id" title="keyword">From</span> <span class="id" title="var">LF</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Maps.html#"><span class="id" title="library">Maps</span></a>.<br/>
<span class="id" title="keyword">From</span> <span class="id" title="var">LF</span> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Imp.html#"><span class="id" title="library">Imp</span></a>.<br/>
</div>

<div class="doc">
Up to now, we've used the more manual part of Coq's tactic
    facilities.  In this chapter, we'll learn more about some of Coq's
    powerful automation features: proof search via the <span class="inlinecode"><span class="id" title="tactic">auto</span></span> tactic,
    automated forward reasoning via the <span class="inlinecode"><span class="id" title="keyword">Ltac</span></span> hypothesis matching
    machinery, and deferred instantiation of existential variables
    using <span class="inlinecode"><span class="id" title="tactic">eapply</span></span> and <span class="inlinecode"><span class="id" title="tactic">eauto</span></span>.  Using these features together with
    Ltac's scripting facilities will enable us to make our proofs
    startlingly short!  Used properly, they can also make proofs more
    maintainable and robust to changes in underlying definitions.  A
    deeper treatment of <span class="inlinecode"><span class="id" title="tactic">auto</span></span> and <span class="inlinecode"><span class="id" title="tactic">eauto</span></span> can be found in the
    <span class="inlinecode"><span class="id" title="var">UseAuto</span></span> chapter in <i>Programming Language Foundations</i>.

<div class="paragraph"> </div>

    There's another major category of automation we haven't discussed
    much yet, namely built-in decision procedures for specific kinds
    of problems: <span class="inlinecode"><span class="id" title="var">lia</span></span> is one example, but there are others.  This
    topic will be deferred for a while longer.

<div class="paragraph"> </div>

    Our motivating example will be this proof, repeated with just a
    few small changes from the <a href="Imp.html"><span class="inlineref">Imp</span></a> chapter.  We will simplify
    this proof in several stages. 
</div>
<div class="code">

<span class="id" title="keyword">Theorem</span> <a id="ceval_deterministic" class="idref" href="#ceval_deterministic"><span class="id" title="lemma">ceval_deterministic</span></a>: <span class="id" title="keyword">∀</span> <a id="c:1" class="idref" href="#c:1"><span class="id" title="binder">c</span></a> <a id="st:2" class="idref" href="#st:2"><span class="id" title="binder">st</span></a> <a id="st<sub>1</sub>:3" class="idref" href="#st<sub>1</sub>:3"><span class="id" title="binder">st<sub>1</sub></span></a> <a id="st<sub>2</sub>:4" class="idref" href="#st<sub>2</sub>:4"><span class="id" title="binder">st<sub>2</sub></span></a>,<br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st:2"><span class="id" title="variable">st</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:1"><span class="id" title="variable">c</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st<sub>1</sub>:3"><span class="id" title="variable">st<sub>1</sub></span></a>  <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st:2"><span class="id" title="variable">st</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:1"><span class="id" title="variable">c</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st<sub>2</sub>:4"><span class="id" title="variable">st<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st<sub>1</sub>:3"><span class="id" title="variable">st<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Auto.html#st<sub>2</sub>:4"><span class="id" title="variable">st<sub>2</sub></span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">c</span> <span class="id" title="var">st</span> <span class="id" title="var">st<sub>1</sub></span> <span class="id" title="var">st<sub>2</sub></span> <span class="id" title="var">E<sub>1</sub></span> <span class="id" title="var">E<sub>2</sub></span>;<br/>
&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> <span class="id" title="tactic">dependent</span> <span class="id" title="var">st<sub>2</sub></span>;<br/>
&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">E<sub>1</sub></span>; <span class="id" title="tactic">intros</span> <span class="id" title="var">st<sub>2</sub></span> <span class="id" title="var">E<sub>2</sub></span>; <span class="id" title="tactic">inversion</span> <span class="id" title="var">E<sub>2</sub></span>; <span class="id" title="tactic">subst</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_Skip&nbsp;*)</span> <span class="id" title="tactic">reflexivity</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_Asgn&nbsp;*)</span> <span class="id" title="tactic">reflexivity</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_Seq&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">IHE1_1</span> <span class="id" title="var">st'0</span> <span class="id" title="var">H<sub>1</sub></span>) <span class="id" title="keyword">in</span> ×.<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IHE1_2</span>. <span class="id" title="tactic">assumption</span>.<br/>
&nbsp;&nbsp;<span class="comment">(*&nbsp;E_IfTrue&nbsp;*)</span><br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;true&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IHE1</span>. <span class="id" title="tactic">assumption</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;false&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>5</sub></span>. <span class="id" title="tactic">discriminate</span>.<br/>
&nbsp;&nbsp;<span class="comment">(*&nbsp;E_IfFalse&nbsp;*)</span><br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;true&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>5</sub></span>. <span class="id" title="tactic">discriminate</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;false&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IHE1</span>. <span class="id" title="tactic">assumption</span>.<br/>
&nbsp;&nbsp;<span class="comment">(*&nbsp;E_WhileFalse&nbsp;*)</span><br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;false&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">reflexivity</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;true&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>2</sub></span>. <span class="id" title="tactic">discriminate</span>.<br/>
&nbsp;&nbsp;<span class="comment">(*&nbsp;E_WhileTrue&nbsp;*)</span><br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;false&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>4</sub></span>. <span class="id" title="tactic">discriminate</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;true&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">IHE1_1</span> <span class="id" title="var">st'0</span> <span class="id" title="var">H<sub>3</sub></span>) <span class="id" title="keyword">in</span> ×.<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">IHE1_2</span>. <span class="id" title="tactic">assumption</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
<a id="lab404"></a><h1 class="section">The <span class="inlinecode"><span class="id" title="tactic">auto</span></span> Tactic</h1>

<div class="paragraph"> </div>

 Thus far, our proof scripts mostly apply relevant hypotheses or
    lemmas by name, and only one at a time. 
</div>
<div class="code">

<span class="id" title="keyword">Example</span> <a id="auto_example_1" class="idref" href="#auto_example_1"><span class="id" title="definition">auto_example_1</span></a> : <span class="id" title="keyword">∀</span> (<a id="P:5" class="idref" href="#P:5"><span class="id" title="binder">P</span></a> <a id="Q:6" class="idref" href="#Q:6"><span class="id" title="binder">Q</span></a> <a id="R:7" class="idref" href="#R:7"><span class="id" title="binder">R</span></a>: <span class="id" title="keyword">Prop</span>),<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#P:5"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#Q:6"><span class="id" title="variable">Q</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#Q:6"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#R:7"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#P:5"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#R:7"><span class="id" title="variable">R</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">P</span> <span class="id" title="var">Q</span> <span class="id" title="var">R</span> <span class="id" title="var">H<sub>1</sub></span> <span class="id" title="var">H<sub>2</sub></span> <span class="id" title="var">H<sub>3</sub></span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <span class="id" title="var">H<sub>2</sub></span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">H<sub>1</sub></span>. <span class="id" title="tactic">assumption</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
The <span class="inlinecode"><span class="id" title="tactic">auto</span></span> tactic frees us from this drudgery by <i>searching</i> for a
    sequence of applications that will prove the goal: 
</div>
<div class="code">

<span class="id" title="keyword">Example</span> <a id="auto_example_1'" class="idref" href="#auto_example_1'"><span class="id" title="definition">auto_example_1'</span></a> : <span class="id" title="keyword">∀</span> (<a id="P:8" class="idref" href="#P:8"><span class="id" title="binder">P</span></a> <a id="Q:9" class="idref" href="#Q:9"><span class="id" title="binder">Q</span></a> <a id="R:10" class="idref" href="#R:10"><span class="id" title="binder">R</span></a>: <span class="id" title="keyword">Prop</span>),<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#P:8"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#Q:9"><span class="id" title="variable">Q</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#Q:9"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#R:10"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#P:8"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#R:10"><span class="id" title="variable">R</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
The <span class="inlinecode"><span class="id" title="tactic">auto</span></span> tactic solves goals that are solvable by any combination of
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" title="tactic">intros</span></span> and

</li>
<li> <span class="inlinecode"><span class="id" title="tactic">apply</span></span> (of hypotheses from the local context, by default). 
</li>
</ul>

<div class="paragraph"> </div>

 Using <span class="inlinecode"><span class="id" title="tactic">auto</span></span> is always "safe" in the sense that it will never fail
    and will never change the proof state: either it completely solves
    the current goal, or it does nothing. 
<div class="paragraph"> </div>

 Here is a larger example showing <span class="inlinecode"><span class="id" title="tactic">auto</span></span>'s power: 
</div>
<div class="code">

<span class="id" title="keyword">Example</span> <a id="auto_example_2" class="idref" href="#auto_example_2"><span class="id" title="definition">auto_example_2</span></a> : <span class="id" title="keyword">∀</span> <a id="P:11" class="idref" href="#P:11"><span class="id" title="binder">P</span></a> <a id="Q:12" class="idref" href="#Q:12"><span class="id" title="binder">Q</span></a> <a id="R:13" class="idref" href="#R:13"><span class="id" title="binder">R</span></a> <a id="S:14" class="idref" href="#S:14"><span class="id" title="binder">S</span></a> <a id="T:15" class="idref" href="#T:15"><span class="id" title="binder">T</span></a> <a id="U:16" class="idref" href="#U:16"><span class="id" title="binder">U</span></a> : <span class="id" title="keyword">Prop</span>,<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#P:11"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#Q:12"><span class="id" title="variable">Q</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#P:11"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#R:13"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#T:15"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#R:13"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#S:14"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#T:15"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#U:16"><span class="id" title="variable">U</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">((</span></a><a class="idref" href="Auto.html#P:11"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#Q:12"><span class="id" title="variable">Q</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#P:11"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#S:14"><span class="id" title="variable">S</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#T:15"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#P:11"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#U:16"><span class="id" title="variable">U</span></a>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">auto</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
Proof search could, in principle, take an arbitrarily long time,
    so there are limits to how far <span class="inlinecode"><span class="id" title="tactic">auto</span></span> will search by default. 
</div>
<div class="code">

<span class="id" title="keyword">Example</span> <a id="auto_example_3" class="idref" href="#auto_example_3"><span class="id" title="definition">auto_example_3</span></a> : <span class="id" title="keyword">∀</span> (<a id="P:17" class="idref" href="#P:17"><span class="id" title="binder">P</span></a> <a id="Q:18" class="idref" href="#Q:18"><span class="id" title="binder">Q</span></a> <a id="R:19" class="idref" href="#R:19"><span class="id" title="binder">R</span></a> <a id="S:20" class="idref" href="#S:20"><span class="id" title="binder">S</span></a> <a id="T:21" class="idref" href="#T:21"><span class="id" title="binder">T</span></a> <a id="U:22" class="idref" href="#U:22"><span class="id" title="binder">U</span></a>: <span class="id" title="keyword">Prop</span>),<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#P:17"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#Q:18"><span class="id" title="variable">Q</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#Q:18"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#R:19"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#R:19"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#S:20"><span class="id" title="variable">S</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#S:20"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#T:21"><span class="id" title="variable">T</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#T:21"><span class="id" title="variable">T</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#U:22"><span class="id" title="variable">U</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#P:17"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#U:22"><span class="id" title="variable">U</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="comment">(*&nbsp;When&nbsp;it&nbsp;cannot&nbsp;solve&nbsp;the&nbsp;goal,&nbsp;<span class="inlinecode"><span class="id" title="tactic">auto</span></span>&nbsp;does&nbsp;nothing&nbsp;*)</span><br/>
&nbsp;&nbsp;<span class="id" title="tactic">auto</span>.<br/>
&nbsp;&nbsp;<span class="comment">(*&nbsp;Optional&nbsp;argument&nbsp;says&nbsp;how&nbsp;deep&nbsp;to&nbsp;search&nbsp;(default&nbsp;is&nbsp;5)&nbsp;*)</span><br/>
&nbsp;&nbsp;<span class="id" title="tactic">auto</span> 6.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
When searching for potential proofs of the current goal,
    <span class="inlinecode"><span class="id" title="tactic">auto</span></span> considers the hypotheses in the current context together
    with a <i>hint database</i> of other lemmas and constructors.  Some
    common lemmas about equality and logical operators are installed
    in this hint database by default. 
</div>
<div class="code">

<span class="id" title="keyword">Example</span> <a id="auto_example_4" class="idref" href="#auto_example_4"><span class="id" title="definition">auto_example_4</span></a> : <span class="id" title="keyword">∀</span> <a id="P:23" class="idref" href="#P:23"><span class="id" title="binder">P</span></a> <a id="Q:24" class="idref" href="#Q:24"><span class="id" title="binder">Q</span></a> <a id="R:25" class="idref" href="#R:25"><span class="id" title="binder">R</span></a> : <span class="id" title="keyword">Prop</span>,<br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#Q:24"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#Q:24"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#R:25"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#P:23"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e<sub>25</sub>"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e<sub>25</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#Q:24"><span class="id" title="variable">Q</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="Auto.html#R:25"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#f031fe1957c4a4a8e217aa46af2b4e<sub>25</sub>"><span class="id" title="notation">)</span></a>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">auto</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
If we want to see which facts <span class="inlinecode"><span class="id" title="tactic">auto</span></span> is using, we can use
    <span class="inlinecode"><span class="id" title="tactic">info_auto</span></span> instead. 
</div>
<div class="code">

<span class="id" title="keyword">Example</span> <a id="auto_example_5" class="idref" href="#auto_example_5"><span class="id" title="definition">auto_example_5</span></a>: 2 <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 2.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">info_auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Example</span> <a id="auto_example_5'" class="idref" href="#auto_example_5'"><span class="id" title="definition">auto_example_5'</span></a> : <span class="id" title="keyword">∀</span> (<a id="P:26" class="idref" href="#P:26"><span class="id" title="binder">P</span></a> <a id="Q:27" class="idref" href="#Q:27"><span class="id" title="binder">Q</span></a> <a id="R:28" class="idref" href="#R:28"><span class="id" title="binder">R</span></a> <a id="S:29" class="idref" href="#S:29"><span class="id" title="binder">S</span></a> <a id="T:30" class="idref" href="#T:30"><span class="id" title="binder">T</span></a> <a id="U:31" class="idref" href="#U:31"><span class="id" title="binder">U</span></a> <a id="W:32" class="idref" href="#W:32"><span class="id" title="binder">W</span></a>: <span class="id" title="keyword">Prop</span>),<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#U:31"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#T:30"><span class="id" title="variable">T</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#W:32"><span class="id" title="variable">W</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#U:31"><span class="id" title="variable">U</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#R:28"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#S:29"><span class="id" title="variable">S</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#S:29"><span class="id" title="variable">S</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#T:30"><span class="id" title="variable">T</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#P:26"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#R:28"><span class="id" title="variable">R</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#U:31"><span class="id" title="variable">U</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#T:30"><span class="id" title="variable">T</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#P:26"><span class="id" title="variable">P</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#T:30"><span class="id" title="variable">T</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">intros</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">info_auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
We can extend the hint database just for the purposes of one
    application of <span class="inlinecode"><span class="id" title="tactic">auto</span></span> by writing "<span class="inlinecode"><span class="id" title="tactic">auto</span></span> <span class="inlinecode"><span class="id" title="keyword">using</span></span> <span class="inlinecode">...</span>". 
</div>
<div class="code">

<span class="id" title="keyword">Lemma</span> <a id="le_antisym" class="idref" href="#le_antisym"><span class="id" title="lemma">le_antisym</span></a> : <span class="id" title="keyword">∀</span> <a id="n:33" class="idref" href="#n:33"><span class="id" title="binder">n</span></a> <a id="m:34" class="idref" href="#m:34"><span class="id" title="binder">m</span></a>: <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#n:33"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a<sub>3</sub>"><span class="id" title="notation">≤</span></a> <a class="idref" href="Auto.html#m:34"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="Auto.html#m:34"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a<sub>3</sub>"><span class="id" title="notation">≤</span></a> <a class="idref" href="Auto.html#n:33"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#n:33"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Auto.html#m:34"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="var">lia</span>. <span class="id" title="keyword">Qed</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Example</span> <a id="auto_example_6" class="idref" href="#auto_example_6"><span class="id" title="definition">auto_example_6</span></a> : <span class="id" title="keyword">∀</span> <a id="n:35" class="idref" href="#n:35"><span class="id" title="binder">n</span></a> <a id="m:36" class="idref" href="#m:36"><span class="id" title="binder">m</span></a> <a id="p:37" class="idref" href="#p:37"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#n:35"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a<sub>3</sub>"><span class="id" title="notation">≤</span></a> <a class="idref" href="Auto.html#p:37"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#n:35"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a<sub>3</sub>"><span class="id" title="notation">≤</span></a> <a class="idref" href="Auto.html#m:36"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="Auto.html#m:36"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a<sub>3</sub>"><span class="id" title="notation">≤</span></a> <a class="idref" href="Auto.html#n:35"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#n:35"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a<sub>3</sub>"><span class="id" title="notation">≤</span></a> <a class="idref" href="Auto.html#p:37"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#n:35"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Auto.html#m:36"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">auto</span> <span class="id" title="keyword">using</span> <a class="idref" href="Auto.html#le_antisym"><span class="id" title="lemma">le_antisym</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
Of course, in any given development there will probably be
    some specific constructors and lemmas that are used very often in
    proofs.  We can add these to the global hint database by writing
<br/>
<span class="inlinecode">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Resolve</span> <span class="id" title="var">T</span> : <span class="id" title="var">core</span>.
</span>    at the top level, where <span class="inlinecode"><span class="id" title="var">T</span></span> is a top-level theorem or a
    constructor of an inductively defined proposition (i.e., anything
    whose type is an implication).  As a shorthand, we can write
<br/>
<span class="inlinecode">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Constructors</span> <span class="id" title="var">c</span> : <span class="id" title="var">core</span>.
</span>    to tell Coq to do a <span class="inlinecode"><span class="id" title="keyword">Hint</span></span> <span class="inlinecode"><span class="id" title="keyword">Resolve</span></span> for <i>all</i> of the constructors
    from the inductive definition of <span class="inlinecode"><span class="id" title="var">c</span></span>.

<div class="paragraph"> </div>

    It is also sometimes necessary to add
<br/>
<span class="inlinecode">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="keyword">Hint</span> <span class="id" title="keyword">Unfold</span> <span class="id" title="var">d</span> : <span class="id" title="var">core</span>.
</span>    where <span class="inlinecode"><span class="id" title="var">d</span></span> is a defined symbol, so that <span class="inlinecode"><span class="id" title="tactic">auto</span></span> knows to expand uses
    of <span class="inlinecode"><span class="id" title="var">d</span></span>, thus enabling further possibilities for applying lemmas that
    it knows about. 
<div class="paragraph"> </div>

 It is also possible to define specialized hint databases (besides
    <span class="inlinecode"><span class="id" title="var">core</span></span>) that can be activated only when needed; indeed, it is good
    style to create your own hint databases instead of polluting
    <span class="inlinecode"><span class="id" title="var">core</span></span>.  See the Coq reference manual for details. 
</div>
<div class="code">

<span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">le_antisym</span> : <span class="id" title="var">core</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Example</span> <a id="auto_example_6'" class="idref" href="#auto_example_6'"><span class="id" title="definition">auto_example_6'</span></a> : <span class="id" title="keyword">∀</span> <a id="n:38" class="idref" href="#n:38"><span class="id" title="binder">n</span></a> <a id="m:39" class="idref" href="#m:39"><span class="id" title="binder">m</span></a> <a id="p:40" class="idref" href="#p:40"><span class="id" title="binder">p</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>,<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#n:38"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a<sub>3</sub>"><span class="id" title="notation">≤</span></a> <a class="idref" href="Auto.html#p:40"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#n:38"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a<sub>3</sub>"><span class="id" title="notation">≤</span></a> <a class="idref" href="Auto.html#m:39"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> <a class="idref" href="Auto.html#m:39"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a<sub>3</sub>"><span class="id" title="notation">≤</span></a> <a class="idref" href="Auto.html#n:38"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">))</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#n:38"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a<sub>3</sub>"><span class="id" title="notation">≤</span></a> <a class="idref" href="Auto.html#p:40"><span class="id" title="variable">p</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#n:38"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Auto.html#m:39"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">auto</span>. <span class="comment">(*&nbsp;picks&nbsp;up&nbsp;hint&nbsp;from&nbsp;database&nbsp;*)</span><br/>
<span class="id" title="keyword">Qed</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Definition</span> <a id="is_fortytwo" class="idref" href="#is_fortytwo"><span class="id" title="definition">is_fortytwo</span></a> <a id="x:41" class="idref" href="#x:41"><span class="id" title="binder">x</span></a> := (<a class="idref" href="Auto.html#x:41"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> 42).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Example</span> <a id="auto_example_7" class="idref" href="#auto_example_7"><span class="id" title="definition">auto_example_7</span></a>: <span class="id" title="keyword">∀</span> <a id="x:42" class="idref" href="#x:42"><span class="id" title="binder">x</span></a>,<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#x:42"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a<sub>3</sub>"><span class="id" title="notation">≤</span></a> 42 <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> 42 <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a<sub>3</sub>"><span class="id" title="notation">≤</span></a> <a class="idref" href="Auto.html#x:42"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#is_fortytwo"><span class="id" title="definition">is_fortytwo</span></a> <a class="idref" href="Auto.html#x:42"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">auto</span>. <span class="comment">(*&nbsp;does&nbsp;nothing&nbsp;*)</span><br/>
<span class="id" title="keyword">Abort</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Hint Unfold</span> <a class="idref" href="Auto.html#is_fortytwo"><span class="id" title="definition">is_fortytwo</span></a> : <span class="id" title="var">core</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Example</span> <a id="auto_example_7'" class="idref" href="#auto_example_7'"><span class="id" title="definition">auto_example_7'</span></a> : <span class="id" title="keyword">∀</span> <a id="x:43" class="idref" href="#x:43"><span class="id" title="binder">x</span></a>,<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#x:43"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a<sub>3</sub>"><span class="id" title="notation">≤</span></a> 42 <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#ba2b0e492d2b4675a0acf3ea92aabadd"><span class="id" title="notation">∧</span></a> 42 <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Peano.html#cb53cf0ee22c036a03b4a9281c68b5a<sub>3</sub>"><span class="id" title="notation">≤</span></a> <a class="idref" href="Auto.html#x:43"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#is_fortytwo"><span class="id" title="definition">is_fortytwo</span></a> <a class="idref" href="Auto.html#x:43"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">auto</span>. <span class="comment">(*&nbsp;try&nbsp;also:&nbsp;info_auto.&nbsp;*)</span><br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
Let's take a first pass over <span class="inlinecode"><span class="id" title="var">ceval_deterministic</span></span> to simplify the
    proof script. 
</div>
<div class="code">

<span class="id" title="keyword">Theorem</span> <a id="ceval_deterministic'" class="idref" href="#ceval_deterministic'"><span class="id" title="lemma">ceval_deterministic'</span></a>: <span class="id" title="keyword">∀</span> <a id="c:44" class="idref" href="#c:44"><span class="id" title="binder">c</span></a> <a id="st:45" class="idref" href="#st:45"><span class="id" title="binder">st</span></a> <a id="st<sub>1</sub>:46" class="idref" href="#st<sub>1</sub>:46"><span class="id" title="binder">st<sub>1</sub></span></a> <a id="st<sub>2</sub>:47" class="idref" href="#st<sub>2</sub>:47"><span class="id" title="binder">st<sub>2</sub></span></a>,<br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st:45"><span class="id" title="variable">st</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:44"><span class="id" title="variable">c</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st<sub>1</sub>:46"><span class="id" title="variable">st<sub>1</sub></span></a>  <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st:45"><span class="id" title="variable">st</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:44"><span class="id" title="variable">c</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st<sub>2</sub>:47"><span class="id" title="variable">st<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st<sub>1</sub>:46"><span class="id" title="variable">st<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Auto.html#st<sub>2</sub>:47"><span class="id" title="variable">st<sub>2</sub></span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">c</span> <span class="id" title="var">st</span> <span class="id" title="var">st<sub>1</sub></span> <span class="id" title="var">st<sub>2</sub></span> <span class="id" title="var">E<sub>1</sub></span> <span class="id" title="var">E<sub>2</sub></span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> <span class="id" title="tactic">dependent</span> <span class="id" title="var">st<sub>2</sub></span>;<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">E<sub>1</sub></span>; <span class="id" title="tactic">intros</span> <span class="id" title="var">st<sub>2</sub></span> <span class="id" title="var">E<sub>2</sub></span>; <span class="id" title="tactic">inversion</span> <span class="id" title="var">E<sub>2</sub></span>; <span class="id" title="tactic">subst</span>; <span class="id" title="tactic">auto</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_Seq&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">IHE1_1</span> <span class="id" title="var">st'0</span> <span class="id" title="var">H<sub>1</sub></span>) <span class="id" title="keyword">in</span> ×.<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">auto</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_IfTrue&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;+ <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;false&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>5</sub></span>. <span class="id" title="tactic">discriminate</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_IfFalse&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;+ <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;true&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>5</sub></span>. <span class="id" title="tactic">discriminate</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_WhileFalse&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;+ <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;true&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>2</sub></span>. <span class="id" title="tactic">discriminate</span>.<br/>
&nbsp;&nbsp;<span class="comment">(*&nbsp;E_WhileTrue&nbsp;*)</span><br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;false&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>4</sub></span>. <span class="id" title="tactic">discriminate</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;true&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">IHE1_1</span> <span class="id" title="var">st'0</span> <span class="id" title="var">H<sub>3</sub></span>) <span class="id" title="keyword">in</span> ×.<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
When we are using a particular tactic many times in a proof, we
    can use a variant of the <span class="inlinecode"><span class="id" title="keyword">Proof</span></span> command to make that tactic into
    a default within the proof.  Saying <span class="inlinecode"><span class="id" title="keyword">Proof</span></span> <span class="inlinecode"><span class="id" title="keyword">with</span></span> <span class="inlinecode"><span class="id" title="var">t</span></span> (where <span class="inlinecode"><span class="id" title="var">t</span></span> is
    an arbitrary tactic) allows us to use <span class="inlinecode"><span class="id" title="var">t<sub>1</sub></span>...</span> as a shorthand for
    <span class="inlinecode"><span class="id" title="var">t<sub>1</sub></span>;<span class="id" title="var">t</span></span> within the proof.  As an illustration, here is an alternate
    version of the previous proof, using <span class="inlinecode"><span class="id" title="keyword">Proof</span></span> <span class="inlinecode"><span class="id" title="keyword">with</span></span> <span class="inlinecode"><span class="id" title="tactic">auto</span></span>. 
</div>
<div class="code">

<span class="id" title="keyword">Theorem</span> <a id="ceval_deterministic'_alt" class="idref" href="#ceval_deterministic'_alt"><span class="id" title="lemma">ceval_deterministic'_alt</span></a>: <span class="id" title="keyword">∀</span> <a id="c:48" class="idref" href="#c:48"><span class="id" title="binder">c</span></a> <a id="st:49" class="idref" href="#st:49"><span class="id" title="binder">st</span></a> <a id="st<sub>1</sub>:50" class="idref" href="#st<sub>1</sub>:50"><span class="id" title="binder">st<sub>1</sub></span></a> <a id="st<sub>2</sub>:51" class="idref" href="#st<sub>2</sub>:51"><span class="id" title="binder">st<sub>2</sub></span></a>,<br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st:49"><span class="id" title="variable">st</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:48"><span class="id" title="variable">c</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st<sub>1</sub>:50"><span class="id" title="variable">st<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st:49"><span class="id" title="variable">st</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:48"><span class="id" title="variable">c</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st<sub>2</sub>:51"><span class="id" title="variable">st<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st<sub>1</sub>:50"><span class="id" title="variable">st<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Auto.html#st<sub>2</sub>:51"><span class="id" title="variable">st<sub>2</sub></span></a>.<br/>
<div class="togglescript" id="proofcontrol1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')"><span class="show"></span></div>
<div class="proofscript" id="proof1" onclick="toggleDisplay('proof1');toggleDisplay('proofcontrol1')">
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">auto</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">c</span> <span class="id" title="var">st</span> <span class="id" title="var">st<sub>1</sub></span> <span class="id" title="var">st<sub>2</sub></span> <span class="id" title="var">E<sub>1</sub></span> <span class="id" title="var">E<sub>2</sub></span>;<br/>
&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> <span class="id" title="tactic">dependent</span> <span class="id" title="var">st<sub>2</sub></span>;<br/>
&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">E<sub>1</sub></span>;<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">st<sub>2</sub></span> <span class="id" title="var">E<sub>2</sub></span>; <span class="id" title="tactic">inversion</span> <span class="id" title="var">E<sub>2</sub></span>; <span class="id" title="tactic">subst</span>...<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_Seq&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">IHE1_1</span> <span class="id" title="var">st'0</span> <span class="id" title="var">H<sub>1</sub></span>) <span class="id" title="keyword">in</span> ×...<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_IfTrue&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;+ <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;false&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>5</sub></span>. <span class="id" title="tactic">discriminate</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_IfFalse&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;+ <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;true&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>5</sub></span>. <span class="id" title="tactic">discriminate</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_WhileFalse&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;+ <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;true&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>2</sub></span>. <span class="id" title="tactic">discriminate</span>.<br/>
&nbsp;&nbsp;<span class="comment">(*&nbsp;E_WhileTrue&nbsp;*)</span><br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;false&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>4</sub></span>. <span class="id" title="tactic">discriminate</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;true&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">IHE1_1</span> <span class="id" title="var">st'0</span> <span class="id" title="var">H<sub>3</sub></span>) <span class="id" title="keyword">in</span> ×...<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>
</div>

<div class="doc">
<a id="lab405"></a><h1 class="section">Searching For Hypotheses</h1>

<div class="paragraph"> </div>

 The proof has become simpler, but there is still an annoying
    amount of repetition. Let's start by tackling the contradiction
    cases. Each of them occurs in a situation where we have both
<br/>
<span class="inlinecode">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">H<sub>1</sub></span>: <span class="id" title="var">beval</span> <span class="id" title="var">st</span> <span class="id" title="var">b</span> = <span class="id" title="var">false</span>
</span>    and
<br/>
<span class="inlinecode">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">H<sub>2</sub></span>: <span class="id" title="var">beval</span> <span class="id" title="var">st</span> <span class="id" title="var">b</span> = <span class="id" title="var">true</span>
</span>    as hypotheses.  The contradiction is evident, but demonstrating it
    is a little complicated: we have to locate the two hypotheses <span class="inlinecode"><span class="id" title="var">H<sub>1</sub></span></span>
    and <span class="inlinecode"><span class="id" title="var">H<sub>2</sub></span></span> and do a <span class="inlinecode"><span class="id" title="tactic">rewrite</span></span> following by a <span class="inlinecode"><span class="id" title="tactic">discriminate</span></span>.  We'd
    like to automate this process.

<div class="paragraph"> </div>

    (In fact, Coq has a built-in tactic <span class="inlinecode"><span class="id" title="tactic">congruence</span></span> that will do the
    job in this case.  But we'll ignore the existence of this tactic
    for now, in order to demonstrate how to build forward search
    tactics by hand.)

<div class="paragraph"> </div>

    As a first step, we can abstract out the piece of script in
    question by writing a little function in Ltac. 
</div>
<div class="code">

<span class="id" title="keyword">Ltac</span> <span class="id" title="var">rwd</span> <span class="id" title="var">H<sub>1</sub></span> <span class="id" title="var">H<sub>2</sub></span> := <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H<sub>1</sub></span> <span class="id" title="keyword">in</span> <span class="id" title="var">H<sub>2</sub></span>; <span class="id" title="tactic">discriminate</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Theorem</span> <a id="ceval_deterministic''" class="idref" href="#ceval_deterministic''"><span class="id" title="lemma">ceval_deterministic''</span></a>: <span class="id" title="keyword">∀</span> <a id="c:52" class="idref" href="#c:52"><span class="id" title="binder">c</span></a> <a id="st:53" class="idref" href="#st:53"><span class="id" title="binder">st</span></a> <a id="st<sub>1</sub>:54" class="idref" href="#st<sub>1</sub>:54"><span class="id" title="binder">st<sub>1</sub></span></a> <a id="st<sub>2</sub>:55" class="idref" href="#st<sub>2</sub>:55"><span class="id" title="binder">st<sub>2</sub></span></a>,<br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st:53"><span class="id" title="variable">st</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:52"><span class="id" title="variable">c</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st<sub>1</sub>:54"><span class="id" title="variable">st<sub>1</sub></span></a>  <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st:53"><span class="id" title="variable">st</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:52"><span class="id" title="variable">c</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st<sub>2</sub>:55"><span class="id" title="variable">st<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st<sub>1</sub>:54"><span class="id" title="variable">st<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Auto.html#st<sub>2</sub>:55"><span class="id" title="variable">st<sub>2</sub></span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">c</span> <span class="id" title="var">st</span> <span class="id" title="var">st<sub>1</sub></span> <span class="id" title="var">st<sub>2</sub></span> <span class="id" title="var">E<sub>1</sub></span> <span class="id" title="var">E<sub>2</sub></span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> <span class="id" title="tactic">dependent</span> <span class="id" title="var">st<sub>2</sub></span>;<br/>
&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">E<sub>1</sub></span>; <span class="id" title="tactic">intros</span> <span class="id" title="var">st<sub>2</sub></span> <span class="id" title="var">E<sub>2</sub></span>; <span class="id" title="tactic">inversion</span> <span class="id" title="var">E<sub>2</sub></span>; <span class="id" title="tactic">subst</span>; <span class="id" title="tactic">auto</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_Seq&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">IHE1_1</span> <span class="id" title="var">st'0</span> <span class="id" title="var">H<sub>1</sub></span>) <span class="id" title="keyword">in</span> ×.<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">auto</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_IfTrue&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;+ <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;false&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">rwd</span> <span class="id" title="var">H</span> <span class="id" title="var">H<sub>5</sub></span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_IfFalse&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;+ <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;true&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">rwd</span> <span class="id" title="var">H</span> <span class="id" title="var">H<sub>5</sub></span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_WhileFalse&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;+ <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;true&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">rwd</span> <span class="id" title="var">H</span> <span class="id" title="var">H<sub>2</sub></span>.<br/>
&nbsp;&nbsp;<span class="comment">(*&nbsp;E_WhileTrue&nbsp;*)</span><br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;false&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">rwd</span> <span class="id" title="var">H</span> <span class="id" title="var">H<sub>4</sub></span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;true&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">IHE1_1</span> <span class="id" title="var">st'0</span> <span class="id" title="var">H<sub>3</sub></span>) <span class="id" title="keyword">in</span> ×.<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">auto</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
That was a bit better, but we really want Coq to discover the
    relevant hypotheses for us.  We can do this by using the <span class="inlinecode"><span class="id" title="keyword">match</span></span>
    <span class="inlinecode"><span class="id" title="keyword">goal</span></span> facility of Ltac. 
</div>
<div class="code">

<span class="id" title="keyword">Ltac</span> <span class="id" title="var">find_rwd</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">H<sub>1</sub></span>: ?<span class="id" title="var">E</span> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Imp.html#:::'true'"><span class="id" title="notation">true</span></a>,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">H<sub>2</sub></span>: ?<span class="id" title="var">E</span> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Imp.html#:::'false'"><span class="id" title="notation">false</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&#x22A2; <span class="id" title="var">_</span> ⇒ <span class="id" title="var">rwd</span> <span class="id" title="var">H<sub>1</sub></span> <span class="id" title="var">H<sub>2</sub></span><br/>
&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
</div>

<div class="doc">
This <span class="inlinecode"><span class="id" title="keyword">match</span></span> <span class="inlinecode"><span class="id" title="keyword">goal</span></span> looks for two distinct hypotheses that
    have the form of equalities, with the same arbitrary expression
    <span class="inlinecode"><span class="id" title="var">E</span></span> on the left and with conflicting boolean values on the right.
    If such hypotheses are found, it binds <span class="inlinecode"><span class="id" title="var">H<sub>1</sub></span></span> and <span class="inlinecode"><span class="id" title="var">H<sub>2</sub></span></span> to their
    names and applies the <span class="inlinecode"><span class="id" title="var">rwd</span></span> tactic to <span class="inlinecode"><span class="id" title="var">H<sub>1</sub></span></span> and <span class="inlinecode"><span class="id" title="var">H<sub>2</sub></span></span>.

<div class="paragraph"> </div>

    Adding this tactic to the ones that we invoke in each case of the
    induction handles all of the contradictory cases. 
</div>
<div class="code">

<span class="id" title="keyword">Theorem</span> <a id="ceval_deterministic'''" class="idref" href="#ceval_deterministic'''"><span class="id" title="lemma">ceval_deterministic'''</span></a>: <span class="id" title="keyword">∀</span> <a id="c:56" class="idref" href="#c:56"><span class="id" title="binder">c</span></a> <a id="st:57" class="idref" href="#st:57"><span class="id" title="binder">st</span></a> <a id="st<sub>1</sub>:58" class="idref" href="#st<sub>1</sub>:58"><span class="id" title="binder">st<sub>1</sub></span></a> <a id="st<sub>2</sub>:59" class="idref" href="#st<sub>2</sub>:59"><span class="id" title="binder">st<sub>2</sub></span></a>,<br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st:57"><span class="id" title="variable">st</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:56"><span class="id" title="variable">c</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st<sub>1</sub>:58"><span class="id" title="variable">st<sub>1</sub></span></a>  <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st:57"><span class="id" title="variable">st</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:56"><span class="id" title="variable">c</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st<sub>2</sub>:59"><span class="id" title="variable">st<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st<sub>1</sub>:58"><span class="id" title="variable">st<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Auto.html#st<sub>2</sub>:59"><span class="id" title="variable">st<sub>2</sub></span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">c</span> <span class="id" title="var">st</span> <span class="id" title="var">st<sub>1</sub></span> <span class="id" title="var">st<sub>2</sub></span> <span class="id" title="var">E<sub>1</sub></span> <span class="id" title="var">E<sub>2</sub></span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> <span class="id" title="tactic">dependent</span> <span class="id" title="var">st<sub>2</sub></span>;<br/>
&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">E<sub>1</sub></span>; <span class="id" title="tactic">intros</span> <span class="id" title="var">st<sub>2</sub></span> <span class="id" title="var">E<sub>2</sub></span>; <span class="id" title="tactic">inversion</span> <span class="id" title="var">E<sub>2</sub></span>; <span class="id" title="tactic">subst</span>; <span class="id" title="tactic">try</span> <span class="id" title="var">find_rwd</span>; <span class="id" title="tactic">auto</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_Seq&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">IHE1_1</span> <span class="id" title="var">st'0</span> <span class="id" title="var">H<sub>1</sub></span>) <span class="id" title="keyword">in</span> ×.<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">auto</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_WhileTrue&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;+ <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;true&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">rewrite</span> (<span class="id" title="var">IHE1_1</span> <span class="id" title="var">st'0</span> <span class="id" title="var">H<sub>3</sub></span>) <span class="id" title="keyword">in</span> ×.<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">auto</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
Let's see about the remaining cases. Each of them involves
    rewriting a hypothesis after feeding it with the required
    condition. We can automate the task of finding the relevant
    hypotheses to rewrite with. 
</div>
<div class="code">

<span class="id" title="keyword">Ltac</span> <span class="id" title="var">find_eqn</span> :=<br/>
&nbsp;&nbsp;<span class="id" title="keyword">match</span> <span class="id" title="keyword">goal</span> <span class="id" title="keyword">with</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">H<sub>1</sub></span>: <span class="id" title="keyword">∀</span> <a id="x:61" class="idref" href="#x:61"><span class="id" title="binder">x</span></a>, ?<span class="id" title="var">P</span> <a class="idref" href="Auto.html#x:60"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> ?<span class="id" title="var">L</span> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> ?<span class="id" title="var">R</span>,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">H<sub>2</sub></span>: ?<span class="id" title="var">P</span> ?<span class="id" title="var">X</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&#x22A2; <span class="id" title="var">_</span> ⇒ <span class="id" title="tactic">rewrite</span> (<span class="id" title="var">H<sub>1</sub></span> <span class="id" title="var">X</span> <span class="id" title="var">H<sub>2</sub></span>) <span class="id" title="keyword">in</span> ×<br/>
&nbsp;&nbsp;<span class="id" title="keyword">end</span>.<br/>
</div>

<div class="doc">
The pattern <span class="inlinecode"><span class="id" title="keyword">∀</span></span> <span class="inlinecode"><span class="id" title="var">x</span>,</span> <span class="inlinecode">?<span class="id" title="var">P</span></span> <span class="inlinecode"><span class="id" title="var">x</span></span> <span class="inlinecode">→</span> <span class="inlinecode">?<span class="id" title="var">L</span></span> <span class="inlinecode">=</span> <span class="inlinecode">?<span class="id" title="var">R</span></span> matches any hypothesis of
    the form "for all <span class="inlinecode"><span class="id" title="var">x</span></span>, <i>some property of <span class="inlinecode"><span class="id" title="var">x</span></span></i> implies <i>some
    equality</i>."  The property of <span class="inlinecode"><span class="id" title="var">x</span></span> is bound to the pattern variable
    <span class="inlinecode"><span class="id" title="var">P</span></span>, and the left- and right-hand sides of the equality are bound
    to <span class="inlinecode"><span class="id" title="var">L</span></span> and <span class="inlinecode"><span class="id" title="var">R</span></span>.  The name of this hypothesis is bound to <span class="inlinecode"><span class="id" title="var">H<sub>1</sub></span></span>.
    Then the pattern <span class="inlinecode">?<span class="id" title="var">P</span></span> <span class="inlinecode">?<span class="id" title="var">X</span></span> matches any hypothesis that provides
    evidence that <span class="inlinecode"><span class="id" title="var">P</span></span> holds for some concrete <span class="inlinecode"><span class="id" title="var">X</span></span>.  If both patterns
    succeed, we apply the <span class="inlinecode"><span class="id" title="tactic">rewrite</span></span> tactic (instantiating the
    quantified <span class="inlinecode"><span class="id" title="var">x</span></span> with <span class="inlinecode"><span class="id" title="var">X</span></span> and providing <span class="inlinecode"><span class="id" title="var">H<sub>2</sub></span></span> as the required
    evidence for <span class="inlinecode"><span class="id" title="var">P</span></span> <span class="inlinecode"><span class="id" title="var">X</span></span>) in all hypotheses and the goal. 
</div>
<div class="code">

<span class="id" title="keyword">Theorem</span> <a id="ceval_deterministic''''" class="idref" href="#ceval_deterministic''''"><span class="id" title="lemma">ceval_deterministic''''</span></a>: <span class="id" title="keyword">∀</span> <a id="c:62" class="idref" href="#c:62"><span class="id" title="binder">c</span></a> <a id="st:63" class="idref" href="#st:63"><span class="id" title="binder">st</span></a> <a id="st<sub>1</sub>:64" class="idref" href="#st<sub>1</sub>:64"><span class="id" title="binder">st<sub>1</sub></span></a> <a id="st<sub>2</sub>:65" class="idref" href="#st<sub>2</sub>:65"><span class="id" title="binder">st<sub>2</sub></span></a>,<br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st:63"><span class="id" title="variable">st</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:62"><span class="id" title="variable">c</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st<sub>1</sub>:64"><span class="id" title="variable">st<sub>1</sub></span></a>  <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st:63"><span class="id" title="variable">st</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:62"><span class="id" title="variable">c</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st<sub>2</sub>:65"><span class="id" title="variable">st<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st<sub>1</sub>:64"><span class="id" title="variable">st<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Auto.html#st<sub>2</sub>:65"><span class="id" title="variable">st<sub>2</sub></span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">c</span> <span class="id" title="var">st</span> <span class="id" title="var">st<sub>1</sub></span> <span class="id" title="var">st<sub>2</sub></span> <span class="id" title="var">E<sub>1</sub></span> <span class="id" title="var">E<sub>2</sub></span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> <span class="id" title="tactic">dependent</span> <span class="id" title="var">st<sub>2</sub></span>;<br/>
&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">E<sub>1</sub></span>; <span class="id" title="tactic">intros</span> <span class="id" title="var">st<sub>2</sub></span> <span class="id" title="var">E<sub>2</sub></span>; <span class="id" title="tactic">inversion</span> <span class="id" title="var">E<sub>2</sub></span>; <span class="id" title="tactic">subst</span>; <span class="id" title="tactic">try</span> <span class="id" title="var">find_rwd</span>;<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">try</span> <span class="id" title="var">find_eqn</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
The big payoff in this approach is that our proof script should be
    more robust in the face of modest changes to our language.  To
    test this, let's try adding a <span class="inlinecode"><span class="id" title="var">REPEAT</span></span> command to the language. 
</div>
<div class="code">

<span class="id" title="keyword">Module</span> <a id="Repeat" class="idref" href="#Repeat"><span class="id" title="module">Repeat</span></a>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Inductive</span> <a id="Repeat.com" class="idref" href="#Repeat.com"><span class="id" title="inductive">com</span></a> : <span class="id" title="keyword">Type</span> :=<br/>
&nbsp;&nbsp;| <a id="Repeat.CSkip" class="idref" href="#Repeat.CSkip"><span class="id" title="constructor">CSkip</span></a><br/>
&nbsp;&nbsp;| <a id="Repeat.CAsgn" class="idref" href="#Repeat.CAsgn"><span class="id" title="constructor">CAsgn</span></a> (<a id="x:68" class="idref" href="#x:68"><span class="id" title="binder">x</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Strings.String.html#string"><span class="id" title="inductive">string</span></a>) (<a id="a:69" class="idref" href="#a:69"><span class="id" title="binder">a</span></a> : <a class="idref" href="Imp.html#aexp"><span class="id" title="inductive">aexp</span></a>)<br/>
&nbsp;&nbsp;| <a id="Repeat.CSeq" class="idref" href="#Repeat.CSeq"><span class="id" title="constructor">CSeq</span></a> (<a id="c<sub>1</sub>:70" class="idref" href="#c<sub>1</sub>:70"><span class="id" title="binder">c<sub>1</sub></span></a> <a id="c<sub>2</sub>:71" class="idref" href="#c<sub>2</sub>:71"><span class="id" title="binder">c<sub>2</sub></span></a> : <a class="idref" href="Auto.html#com:66"><span class="id" title="inductive">com</span></a>)<br/>
&nbsp;&nbsp;| <a id="Repeat.CIf" class="idref" href="#Repeat.CIf"><span class="id" title="constructor">CIf</span></a> (<a id="b:72" class="idref" href="#b:72"><span class="id" title="binder">b</span></a> : <a class="idref" href="Imp.html#bexp"><span class="id" title="inductive">bexp</span></a>) (<a id="c<sub>1</sub>:73" class="idref" href="#c<sub>1</sub>:73"><span class="id" title="binder">c<sub>1</sub></span></a> <a id="c<sub>2</sub>:74" class="idref" href="#c<sub>2</sub>:74"><span class="id" title="binder">c<sub>2</sub></span></a> : <a class="idref" href="Auto.html#com:66"><span class="id" title="inductive">com</span></a>)<br/>
&nbsp;&nbsp;| <a id="Repeat.CWhile" class="idref" href="#Repeat.CWhile"><span class="id" title="constructor">CWhile</span></a> (<a id="b:75" class="idref" href="#b:75"><span class="id" title="binder">b</span></a> : <a class="idref" href="Imp.html#bexp"><span class="id" title="inductive">bexp</span></a>) (<a id="c:76" class="idref" href="#c:76"><span class="id" title="binder">c</span></a> : <a class="idref" href="Auto.html#com:66"><span class="id" title="inductive">com</span></a>)<br/>
&nbsp;&nbsp;| <a id="Repeat.CRepeat" class="idref" href="#Repeat.CRepeat"><span class="id" title="constructor">CRepeat</span></a> (<a id="c:77" class="idref" href="#c:77"><span class="id" title="binder">c</span></a> : <a class="idref" href="Auto.html#com:66"><span class="id" title="inductive">com</span></a>) (<a id="b:78" class="idref" href="#b:78"><span class="id" title="binder">b</span></a> : <a class="idref" href="Imp.html#bexp"><span class="id" title="inductive">bexp</span></a>).<br/>
</div>

<div class="doc">
<span class="inlinecode"><span class="id" title="var">REPEAT</span></span> behaves like <span class="inlinecode"><span class="id" title="var">while</span></span>, except that the loop guard is
    checked <i>after</i> each execution of the body, with the loop
    repeating as long as the guard stays <i>false</i>.  Because of this,
    the body will always execute at least once. 
</div>
<div class="code">

<span class="id" title="keyword">Notation</span> <a id="Repeat.:com::'repeat'_x_'until'_x_'end'" class="idref" href="#Repeat.:com::'repeat'_x_'until'_x_'end'"><span class="id" title="notation">&quot;</span></a>'repeat' x 'until' y 'end'" :=<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="Auto.html#Repeat.CRepeat"><span class="id" title="constructor">CRepeat</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>)<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">com</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">x</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99).<br/>
<span class="id" title="keyword">Notation</span> <a id="Repeat.:com::'skip'" class="idref" href="#Repeat.:com::'skip'"><span class="id" title="notation">&quot;</span></a>'skip'"  :=<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#Repeat.CSkip"><span class="id" title="constructor">CSkip</span></a> (<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">com</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0).<br/>
<span class="id" title="keyword">Notation</span> <a id="c44f2be0d94374087d3aa3c10eb1cd3f" class="idref" href="#c44f2be0d94374087d3aa3c10eb1cd3f"><span class="id" title="notation">&quot;</span></a>x := y"  :=<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="Auto.html#Repeat.CAsgn"><span class="id" title="constructor">CAsgn</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>)<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">com</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">x</span> <span class="id" title="keyword">constr</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 85, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>).<br/>
<span class="id" title="keyword">Notation</span> <a id="ff9263be1199b3ea0f6b140fe65590d<sub>2</sub>" class="idref" href="#ff9263be1199b3ea0f6b140fe65590d<sub>2</sub>"><span class="id" title="notation">&quot;</span></a>x ; y" :=<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="Auto.html#Repeat.CSeq"><span class="id" title="constructor">CSeq</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>)<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">com</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 90, <span class="id" title="tactic">right</span> <span class="id" title="keyword">associativity</span>).<br/>
<span class="id" title="keyword">Notation</span> <a id="Repeat.:com::'if'_x_'then'_x_'else'_x_'end'" class="idref" href="#Repeat.:com::'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">&quot;</span></a>'if' x 'then' y 'else' z 'end'" :=<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="Auto.html#Repeat.CIf"><span class="id" title="constructor">CIf</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">z</span>)<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">com</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 89, <span class="id" title="var">x</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">z</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99).<br/>
<span class="id" title="keyword">Notation</span> <a id="Repeat.:com::'while'_x_'do'_x_'end'" class="idref" href="#Repeat.:com::'while'_x_'do'_x_'end'"><span class="id" title="notation">&quot;</span></a>'while' x 'do' y 'end'" :=<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<a class="idref" href="Auto.html#Repeat.CWhile"><span class="id" title="constructor">CWhile</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>)<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="keyword">in</span> <span class="id" title="var">custom</span> <span class="id" title="var">com</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 89, <span class="id" title="var">x</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">y</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Reserved Notation</span> &quot;st '=[' c ']=&gt;' st'"<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 40, <span class="id" title="var">c</span> <span class="id" title="var">custom</span> <span class="id" title="var">com</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 99, <span class="id" title="var">st'</span> <span class="id" title="keyword">constr</span> <span class="id" title="tactic">at</span> <span class="id" title="var">next</span> <span class="id" title="keyword">level</span>).<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Inductive</span> <a id="Repeat.ceval" class="idref" href="#Repeat.ceval"><span class="id" title="inductive">ceval</span></a> : <a class="idref" href="Auto.html#Repeat.com"><span class="id" title="inductive">com</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Imp.html#state"><span class="id" title="definition">state</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Imp.html#state"><span class="id" title="definition">state</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span> :=<br/>
&nbsp;&nbsp;| <a id="Repeat.E_Skip" class="idref" href="#Repeat.E_Skip"><span class="id" title="constructor">E_Skip</span></a> : <span class="id" title="keyword">∀</span> <a id="st:81" class="idref" href="#st:81"><span class="id" title="binder">st</span></a>,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st:81"><span class="id" title="variable">st</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#Repeat.:com::'skip'"><span class="id" title="notation">skip</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st:81"><span class="id" title="variable">st</span></a><br/>
&nbsp;&nbsp;| <a id="Repeat.E_Asgn" class="idref" href="#Repeat.E_Asgn"><span class="id" title="constructor">E_Asgn</span></a>  : <span class="id" title="keyword">∀</span> <a id="st:82" class="idref" href="#st:82"><span class="id" title="binder">st</span></a> <a id="a<sub>1</sub>:83" class="idref" href="#a<sub>1</sub>:83"><span class="id" title="binder">a<sub>1</sub></span></a> <a id="n:84" class="idref" href="#n:84"><span class="id" title="binder">n</span></a> <a id="x:85" class="idref" href="#x:85"><span class="id" title="binder">x</span></a>,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#aeval"><span class="id" title="definition">aeval</span></a> <a class="idref" href="Auto.html#st:82"><span class="id" title="variable">st</span></a> <a class="idref" href="Auto.html#a<sub>1</sub>:83"><span class="id" title="variable">a<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Auto.html#n:84"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st:82"><span class="id" title="variable">st</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#x:85"><span class="id" title="variable">x</span></a> <a class="idref" href="Auto.html#c44f2be0d94374087d3aa3c10eb1cd3f"><span class="id" title="notation">:=</span></a> <a class="idref" href="Auto.html#a<sub>1</sub>:83"><span class="id" title="variable">a<sub>1</sub></span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="Auto.html#x:85"><span class="id" title="variable">x</span></a> <a class="idref" href="Maps.html#630986d105c0f1782d085b9a306379a<sub>7</sub>"><span class="id" title="notation">!<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>&gt;</span></span></span></a> <a class="idref" href="Auto.html#n:84"><span class="id" title="variable">n</span></a> <a class="idref" href="Maps.html#630986d105c0f1782d085b9a306379a<sub>7</sub>"><span class="id" title="notation">;</span></a> <a class="idref" href="Auto.html#st:82"><span class="id" title="variable">st</span></a><a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">)</span></a><br/>
&nbsp;&nbsp;| <a id="Repeat.E_Seq" class="idref" href="#Repeat.E_Seq"><span class="id" title="constructor">E_Seq</span></a> : <span class="id" title="keyword">∀</span> <a id="c<sub>1</sub>:86" class="idref" href="#c<sub>1</sub>:86"><span class="id" title="binder">c<sub>1</sub></span></a> <a id="c<sub>2</sub>:87" class="idref" href="#c<sub>2</sub>:87"><span class="id" title="binder">c<sub>2</sub></span></a> <a id="st:88" class="idref" href="#st:88"><span class="id" title="binder">st</span></a> <a id="st':89" class="idref" href="#st':89"><span class="id" title="binder">st'</span></a> <a id="st'':90" class="idref" href="#st'':90"><span class="id" title="binder">st''</span></a>,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st:88"><span class="id" title="variable">st</span></a>  <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c<sub>1</sub>:86"><span class="id" title="variable">c<sub>1</sub></span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st':89"><span class="id" title="variable">st'</span></a>  <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st':89"><span class="id" title="variable">st'</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c<sub>2</sub>:87"><span class="id" title="variable">c<sub>2</sub></span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st'':90"><span class="id" title="variable">st''</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st:88"><span class="id" title="variable">st</span></a>  <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c<sub>1</sub>:86"><span class="id" title="variable">c<sub>1</sub></span></a> <a class="idref" href="Auto.html#ff9263be1199b3ea0f6b140fe65590d<sub>2</sub>"><span class="id" title="notation">;</span></a> <a class="idref" href="Auto.html#c<sub>2</sub>:87"><span class="id" title="variable">c<sub>2</sub></span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st'':90"><span class="id" title="variable">st''</span></a><br/>
&nbsp;&nbsp;| <a id="Repeat.E_IfTrue" class="idref" href="#Repeat.E_IfTrue"><span class="id" title="constructor">E_IfTrue</span></a> : <span class="id" title="keyword">∀</span> <a id="st:91" class="idref" href="#st:91"><span class="id" title="binder">st</span></a> <a id="st':92" class="idref" href="#st':92"><span class="id" title="binder">st'</span></a> <a id="b:93" class="idref" href="#b:93"><span class="id" title="binder">b</span></a> <a id="c<sub>1</sub>:94" class="idref" href="#c<sub>1</sub>:94"><span class="id" title="binder">c<sub>1</sub></span></a> <a id="c<sub>2</sub>:95" class="idref" href="#c<sub>2</sub>:95"><span class="id" title="binder">c<sub>2</sub></span></a>,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#beval"><span class="id" title="definition">beval</span></a> <a class="idref" href="Auto.html#st:91"><span class="id" title="variable">st</span></a> <a class="idref" href="Auto.html#b:93"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Imp.html#:::'true'"><span class="id" title="notation">true</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st:91"><span class="id" title="variable">st</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c<sub>1</sub>:94"><span class="id" title="variable">c<sub>1</sub></span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st':92"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st:91"><span class="id" title="variable">st</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#Repeat.:com::'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">if</span></a> <a class="idref" href="Auto.html#b:93"><span class="id" title="variable">b</span></a> <a class="idref" href="Auto.html#Repeat.:com::'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">then</span></a> <a class="idref" href="Auto.html#c<sub>1</sub>:94"><span class="id" title="variable">c<sub>1</sub></span></a> <a class="idref" href="Auto.html#Repeat.:com::'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">else</span></a> <a class="idref" href="Auto.html#c<sub>2</sub>:95"><span class="id" title="variable">c<sub>2</sub></span></a> <a class="idref" href="Auto.html#Repeat.:com::'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">end</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st':92"><span class="id" title="variable">st'</span></a><br/>
&nbsp;&nbsp;| <a id="Repeat.E_IfFalse" class="idref" href="#Repeat.E_IfFalse"><span class="id" title="constructor">E_IfFalse</span></a> : <span class="id" title="keyword">∀</span> <a id="st:96" class="idref" href="#st:96"><span class="id" title="binder">st</span></a> <a id="st':97" class="idref" href="#st':97"><span class="id" title="binder">st'</span></a> <a id="b:98" class="idref" href="#b:98"><span class="id" title="binder">b</span></a> <a id="c<sub>1</sub>:99" class="idref" href="#c<sub>1</sub>:99"><span class="id" title="binder">c<sub>1</sub></span></a> <a id="c<sub>2</sub>:100" class="idref" href="#c<sub>2</sub>:100"><span class="id" title="binder">c<sub>2</sub></span></a>,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#beval"><span class="id" title="definition">beval</span></a> <a class="idref" href="Auto.html#st:96"><span class="id" title="variable">st</span></a> <a class="idref" href="Auto.html#b:98"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Imp.html#:::'false'"><span class="id" title="notation">false</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st:96"><span class="id" title="variable">st</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c<sub>2</sub>:100"><span class="id" title="variable">c<sub>2</sub></span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st':97"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st:96"><span class="id" title="variable">st</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#Repeat.:com::'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">if</span></a> <a class="idref" href="Auto.html#b:98"><span class="id" title="variable">b</span></a> <a class="idref" href="Auto.html#Repeat.:com::'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">then</span></a> <a class="idref" href="Auto.html#c<sub>1</sub>:99"><span class="id" title="variable">c<sub>1</sub></span></a> <a class="idref" href="Auto.html#Repeat.:com::'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">else</span></a> <a class="idref" href="Auto.html#c<sub>2</sub>:100"><span class="id" title="variable">c<sub>2</sub></span></a> <a class="idref" href="Auto.html#Repeat.:com::'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">end</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st':97"><span class="id" title="variable">st'</span></a><br/>
&nbsp;&nbsp;| <a id="Repeat.E_WhileFalse" class="idref" href="#Repeat.E_WhileFalse"><span class="id" title="constructor">E_WhileFalse</span></a> : <span class="id" title="keyword">∀</span> <a id="b:101" class="idref" href="#b:101"><span class="id" title="binder">b</span></a> <a id="st:102" class="idref" href="#st:102"><span class="id" title="binder">st</span></a> <a id="c:103" class="idref" href="#c:103"><span class="id" title="binder">c</span></a>,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#beval"><span class="id" title="definition">beval</span></a> <a class="idref" href="Auto.html#st:102"><span class="id" title="variable">st</span></a> <a class="idref" href="Auto.html#b:101"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Imp.html#:::'false'"><span class="id" title="notation">false</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st:102"><span class="id" title="variable">st</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#Repeat.:com::'while'_x_'do'_x_'end'"><span class="id" title="notation">while</span></a> <a class="idref" href="Auto.html#b:101"><span class="id" title="variable">b</span></a> <a class="idref" href="Auto.html#Repeat.:com::'while'_x_'do'_x_'end'"><span class="id" title="notation">do</span></a> <a class="idref" href="Auto.html#c:103"><span class="id" title="variable">c</span></a> <a class="idref" href="Auto.html#Repeat.:com::'while'_x_'do'_x_'end'"><span class="id" title="notation">end</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st:102"><span class="id" title="variable">st</span></a><br/>
&nbsp;&nbsp;| <a id="Repeat.E_WhileTrue" class="idref" href="#Repeat.E_WhileTrue"><span class="id" title="constructor">E_WhileTrue</span></a> : <span class="id" title="keyword">∀</span> <a id="st:104" class="idref" href="#st:104"><span class="id" title="binder">st</span></a> <a id="st':105" class="idref" href="#st':105"><span class="id" title="binder">st'</span></a> <a id="st'':106" class="idref" href="#st'':106"><span class="id" title="binder">st''</span></a> <a id="b:107" class="idref" href="#b:107"><span class="id" title="binder">b</span></a> <a id="c:108" class="idref" href="#c:108"><span class="id" title="binder">c</span></a>,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#beval"><span class="id" title="definition">beval</span></a> <a class="idref" href="Auto.html#st:104"><span class="id" title="variable">st</span></a> <a class="idref" href="Auto.html#b:107"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Imp.html#:::'true'"><span class="id" title="notation">true</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st:104"><span class="id" title="variable">st</span></a>  <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:108"><span class="id" title="variable">c</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st':105"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st':105"><span class="id" title="variable">st'</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#Repeat.:com::'while'_x_'do'_x_'end'"><span class="id" title="notation">while</span></a> <a class="idref" href="Auto.html#b:107"><span class="id" title="variable">b</span></a> <a class="idref" href="Auto.html#Repeat.:com::'while'_x_'do'_x_'end'"><span class="id" title="notation">do</span></a> <a class="idref" href="Auto.html#c:108"><span class="id" title="variable">c</span></a> <a class="idref" href="Auto.html#Repeat.:com::'while'_x_'do'_x_'end'"><span class="id" title="notation">end</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st'':106"><span class="id" title="variable">st''</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st:104"><span class="id" title="variable">st</span></a>  <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#Repeat.:com::'while'_x_'do'_x_'end'"><span class="id" title="notation">while</span></a> <a class="idref" href="Auto.html#b:107"><span class="id" title="variable">b</span></a> <a class="idref" href="Auto.html#Repeat.:com::'while'_x_'do'_x_'end'"><span class="id" title="notation">do</span></a> <a class="idref" href="Auto.html#c:108"><span class="id" title="variable">c</span></a> <a class="idref" href="Auto.html#Repeat.:com::'while'_x_'do'_x_'end'"><span class="id" title="notation">end</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st'':106"><span class="id" title="variable">st''</span></a><br/>
&nbsp;&nbsp;| <a id="Repeat.E_RepeatEnd" class="idref" href="#Repeat.E_RepeatEnd"><span class="id" title="constructor">E_RepeatEnd</span></a> : <span class="id" title="keyword">∀</span> <a id="st:109" class="idref" href="#st:109"><span class="id" title="binder">st</span></a> <a id="st':110" class="idref" href="#st':110"><span class="id" title="binder">st'</span></a> <a id="b:111" class="idref" href="#b:111"><span class="id" title="binder">b</span></a> <a id="c:112" class="idref" href="#c:112"><span class="id" title="binder">c</span></a>,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st:109"><span class="id" title="variable">st</span></a>  <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:112"><span class="id" title="variable">c</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st':110"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#beval"><span class="id" title="definition">beval</span></a> <a class="idref" href="Auto.html#st':110"><span class="id" title="variable">st'</span></a> <a class="idref" href="Auto.html#b:111"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Imp.html#:::'true'"><span class="id" title="notation">true</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st:109"><span class="id" title="variable">st</span></a>  <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#Repeat.:com::'repeat'_x_'until'_x_'end'"><span class="id" title="notation">repeat</span></a> <a class="idref" href="Auto.html#c:112"><span class="id" title="variable">c</span></a> <a class="idref" href="Auto.html#Repeat.:com::'repeat'_x_'until'_x_'end'"><span class="id" title="notation">until</span></a> <a class="idref" href="Auto.html#b:111"><span class="id" title="variable">b</span></a> <a class="idref" href="Auto.html#Repeat.:com::'repeat'_x_'until'_x_'end'"><span class="id" title="notation">end</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st':110"><span class="id" title="variable">st'</span></a><br/>
&nbsp;&nbsp;| <a id="Repeat.E_RepeatLoop" class="idref" href="#Repeat.E_RepeatLoop"><span class="id" title="constructor">E_RepeatLoop</span></a> : <span class="id" title="keyword">∀</span> <a id="st:113" class="idref" href="#st:113"><span class="id" title="binder">st</span></a> <a id="st':114" class="idref" href="#st':114"><span class="id" title="binder">st'</span></a> <a id="st'':115" class="idref" href="#st'':115"><span class="id" title="binder">st''</span></a> <a id="b:116" class="idref" href="#b:116"><span class="id" title="binder">b</span></a> <a id="c:117" class="idref" href="#c:117"><span class="id" title="binder">c</span></a>,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st:113"><span class="id" title="variable">st</span></a>  <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:117"><span class="id" title="variable">c</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st':114"><span class="id" title="variable">st'</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#beval"><span class="id" title="definition">beval</span></a> <a class="idref" href="Auto.html#st':114"><span class="id" title="variable">st'</span></a> <a class="idref" href="Auto.html#b:116"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Imp.html#:::'false'"><span class="id" title="notation">false</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st':114"><span class="id" title="variable">st'</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#Repeat.:com::'repeat'_x_'until'_x_'end'"><span class="id" title="notation">repeat</span></a> <a class="idref" href="Auto.html#c:117"><span class="id" title="variable">c</span></a> <a class="idref" href="Auto.html#Repeat.:com::'repeat'_x_'until'_x_'end'"><span class="id" title="notation">until</span></a> <a class="idref" href="Auto.html#b:116"><span class="id" title="variable">b</span></a> <a class="idref" href="Auto.html#Repeat.:com::'repeat'_x_'until'_x_'end'"><span class="id" title="notation">end</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st'':115"><span class="id" title="variable">st''</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Auto.html#st:113"><span class="id" title="variable">st</span></a>  <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#Repeat.:com::'repeat'_x_'until'_x_'end'"><span class="id" title="notation">repeat</span></a> <a class="idref" href="Auto.html#c:117"><span class="id" title="variable">c</span></a> <a class="idref" href="Auto.html#Repeat.:com::'repeat'_x_'until'_x_'end'"><span class="id" title="notation">until</span></a> <a class="idref" href="Auto.html#b:116"><span class="id" title="variable">b</span></a> <a class="idref" href="Auto.html#Repeat.:com::'repeat'_x_'until'_x_'end'"><span class="id" title="notation">end</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st'':115"><span class="id" title="variable">st''</span></a><br/>
<br/>
&nbsp;&nbsp;<span class="id" title="keyword">where</span> <a id="029586011588c831225d1548c37559d<sub>6</sub>" class="idref" href="#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">&quot;</span></a>st =[ c ]=&gt; st'" := (<a class="idref" href="Auto.html#ceval:80"><span class="id" title="inductive">ceval</span></a> <span class="id" title="var">c</span> <span class="id" title="var">st</span> <span class="id" title="var">st'</span>).<br/>
</div>

<div class="doc">
Our first attempt at the determinacy proof does not quite succeed:
    the <span class="inlinecode"><span class="id" title="var">E_RepeatEnd</span></span> and <span class="inlinecode"><span class="id" title="var">E_RepeatLoop</span></span> cases are not handled by our
    previous automation. 
</div>
<div class="code">

<span class="id" title="keyword">Theorem</span> <a id="Repeat.ceval_deterministic" class="idref" href="#Repeat.ceval_deterministic"><span class="id" title="lemma">ceval_deterministic</span></a>: <span class="id" title="keyword">∀</span> <a id="c:118" class="idref" href="#c:118"><span class="id" title="binder">c</span></a> <a id="st:119" class="idref" href="#st:119"><span class="id" title="binder">st</span></a> <a id="st<sub>1</sub>:120" class="idref" href="#st<sub>1</sub>:120"><span class="id" title="binder">st<sub>1</sub></span></a> <a id="st<sub>2</sub>:121" class="idref" href="#st<sub>2</sub>:121"><span class="id" title="binder">st<sub>2</sub></span></a>,<br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st:119"><span class="id" title="variable">st</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:118"><span class="id" title="variable">c</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st<sub>1</sub>:120"><span class="id" title="variable">st<sub>1</sub></span></a>  <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st:119"><span class="id" title="variable">st</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:118"><span class="id" title="variable">c</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st<sub>2</sub>:121"><span class="id" title="variable">st<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st<sub>1</sub>:120"><span class="id" title="variable">st<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Auto.html#st<sub>2</sub>:121"><span class="id" title="variable">st<sub>2</sub></span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">c</span> <span class="id" title="var">st</span> <span class="id" title="var">st<sub>1</sub></span> <span class="id" title="var">st<sub>2</sub></span> <span class="id" title="var">E<sub>1</sub></span> <span class="id" title="var">E<sub>2</sub></span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> <span class="id" title="tactic">dependent</span> <span class="id" title="var">st<sub>2</sub></span>;<br/>
&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">E<sub>1</sub></span>;<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">st<sub>2</sub></span> <span class="id" title="var">E<sub>2</sub></span>; <span class="id" title="tactic">inversion</span> <span class="id" title="var">E<sub>2</sub></span>; <span class="id" title="tactic">subst</span>; <span class="id" title="tactic">try</span> <span class="id" title="var">find_rwd</span>; <span class="id" title="tactic">try</span> <span class="id" title="var">find_eqn</span>; <span class="id" title="tactic">auto</span>.<br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_RepeatEnd&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;+ <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;false&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">find_rwd</span>.<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="comment">(*&nbsp;oops:&nbsp;why&nbsp;didn't&nbsp;<span class="inlinecode"><span class="id" title="var">find_rwd</span></span>&nbsp;solve&nbsp;this&nbsp;for&nbsp;us&nbsp;already?<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;answer:&nbsp;we&nbsp;did&nbsp;things&nbsp;in&nbsp;the&nbsp;wrong&nbsp;order.&nbsp;*)</span><br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;E_RepeatLoop&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;+ <span class="comment">(*&nbsp;b&nbsp;evaluates&nbsp;to&nbsp;true&nbsp;(contradiction)&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">find_rwd</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
Fortunately, to fix this, we just have to swap the invocations of
    <span class="inlinecode"><span class="id" title="var">find_eqn</span></span> and <span class="inlinecode"><span class="id" title="var">find_rwd</span></span>. 
</div>
<div class="code">

<span class="id" title="keyword">Theorem</span> <a id="Repeat.ceval_deterministic'" class="idref" href="#Repeat.ceval_deterministic'"><span class="id" title="lemma">ceval_deterministic'</span></a>: <span class="id" title="keyword">∀</span> <a id="c:122" class="idref" href="#c:122"><span class="id" title="binder">c</span></a> <a id="st:123" class="idref" href="#st:123"><span class="id" title="binder">st</span></a> <a id="st<sub>1</sub>:124" class="idref" href="#st<sub>1</sub>:124"><span class="id" title="binder">st<sub>1</sub></span></a> <a id="st<sub>2</sub>:125" class="idref" href="#st<sub>2</sub>:125"><span class="id" title="binder">st<sub>2</sub></span></a>,<br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st:123"><span class="id" title="variable">st</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:122"><span class="id" title="variable">c</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st<sub>1</sub>:124"><span class="id" title="variable">st<sub>1</sub></span></a>  <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st:123"><span class="id" title="variable">st</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">=[</span></a> <a class="idref" href="Auto.html#c:122"><span class="id" title="variable">c</span></a> <a class="idref" href="Auto.html#029586011588c831225d1548c37559d<sub>6</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#st<sub>2</sub>:125"><span class="id" title="variable">st<sub>2</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#st<sub>1</sub>:124"><span class="id" title="variable">st<sub>1</sub></span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#6cd0f7b28b6092304087c7049437bb1a"><span class="id" title="notation">=</span></a> <a class="idref" href="Auto.html#st<sub>2</sub>:125"><span class="id" title="variable">st<sub>2</sub></span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">c</span> <span class="id" title="var">st</span> <span class="id" title="var">st<sub>1</sub></span> <span class="id" title="var">st<sub>2</sub></span> <span class="id" title="var">E<sub>1</sub></span> <span class="id" title="var">E<sub>2</sub></span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">generalize</span> <span class="id" title="tactic">dependent</span> <span class="id" title="var">st<sub>2</sub></span>;<br/>
&nbsp;&nbsp;<span class="id" title="tactic">induction</span> <span class="id" title="var">E<sub>1</sub></span>;<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">st<sub>2</sub></span> <span class="id" title="var">E<sub>2</sub></span>; <span class="id" title="tactic">inversion</span> <span class="id" title="var">E<sub>2</sub></span>; <span class="id" title="tactic">subst</span>; <span class="id" title="tactic">try</span> <span class="id" title="var">find_eqn</span>; <span class="id" title="tactic">try</span> <span class="id" title="var">find_rwd</span>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">End</span> <a class="idref" href="Auto.html#Repeat"><span class="id" title="module">Repeat</span></a>.<br/>
</div>

<div class="doc">
These examples just give a flavor of what "hyper-automation"
    can achieve in Coq.  The details of <span class="inlinecode"><span class="id" title="keyword">match</span></span> <span class="inlinecode"><span class="id" title="keyword">goal</span></span> are a bit
    tricky (and debugging scripts using it is, frankly, not very
    pleasant).  But it is well worth adding at least simple uses to
    your proofs, both to avoid tedium and to "future proof" them. 
<div class="paragraph"> </div>

<a id="lab406"></a><h1 class="section">Tactics <span class="inlinecode"><span class="id" title="tactic">eapply</span></span> and <span class="inlinecode"><span class="id" title="tactic">eauto</span></span></h1>

<div class="paragraph"> </div>

 To close the chapter, we'll introduce one more convenient feature
    of Coq: its ability to delay instantiation of quantifiers.  To
    motivate this feature, recall this example from the <a href="Imp.html"><span class="inlineref">Imp</span></a>
    chapter: 
</div>
<div class="code">

<span class="id" title="keyword">Example</span> <a id="ceval_example1" class="idref" href="#ceval_example1"><span class="id" title="definition">ceval_example1</span></a>:<br/>
&nbsp;&nbsp;<a class="idref" href="Imp.html#empty_st"><span class="id" title="definition">empty_st</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">=[</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#X"><span class="id" title="definition">X</span></a> <a class="idref" href="Imp.html#91e9aa710642047a93142bdf557f1a1b"><span class="id" title="notation">:=</span></a> 2<a class="idref" href="Imp.html#313afe74ec81f2da17d8e7bca3b042e<sub>7</sub>"><span class="id" title="notation">;</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#:com:com_scope:'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">if</span></a> <a class="idref" href="Imp.html#5a96623d056c293ebbcdf03005796e<sub>09</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="Imp.html#X"><span class="id" title="definition">X</span></a> <a class="idref" href="Imp.html#d43f15edd52d29ae8e0f90730a25ed5c"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="Imp.html#5a96623d056c293ebbcdf03005796e<sub>09</sub>"><span class="id" title="notation">)</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#:com:com_scope:'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">then</span></a> <a class="idref" href="Imp.html#Y"><span class="id" title="definition">Y</span></a> <a class="idref" href="Imp.html#91e9aa710642047a93142bdf557f1a1b"><span class="id" title="notation">:=</span></a> 3<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#:com:com_scope:'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">else</span></a> <a class="idref" href="Imp.html#Z"><span class="id" title="definition">Z</span></a> <a class="idref" href="Imp.html#91e9aa710642047a93142bdf557f1a1b"><span class="id" title="notation">:=</span></a> 4<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#:com:com_scope:'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">end</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="Imp.html#Z"><span class="id" title="definition">Z</span></a> <a class="idref" href="Maps.html#630986d105c0f1782d085b9a306379a<sub>7</sub>"><span class="id" title="notation">!<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>&gt;</span></span></span></a> 4 <a class="idref" href="Maps.html#630986d105c0f1782d085b9a306379a<sub>7</sub>"><span class="id" title="notation">;</span></a> <a class="idref" href="Imp.html#X"><span class="id" title="definition">X</span></a> <a class="idref" href="Imp.html#c78e6ed7f05f8146dc2bcb368255bece"><span class="id" title="notation">!<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>&gt;</span></span></span></a> 2<a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">)</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="comment">(*&nbsp;We&nbsp;supply&nbsp;the&nbsp;intermediate&nbsp;state&nbsp;<span class="inlinecode"><span class="id" title="var">st'</span></span>...&nbsp;*)</span><br/>
&nbsp;&nbsp;<span class="id" title="tactic">apply</span> <a class="idref" href="Imp.html#E_Seq"><span class="id" title="constructor">E_Seq</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="Imp.html#X"><span class="id" title="definition">X</span></a> <a class="idref" href="Imp.html#c78e6ed7f05f8146dc2bcb368255bece"><span class="id" title="notation">!<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>&gt;</span></span></span></a> 2).<br/>
&nbsp;&nbsp;- <span class="id" title="tactic">apply</span> <a class="idref" href="Imp.html#E_Asgn"><span class="id" title="constructor">E_Asgn</span></a>. <span class="id" title="tactic">reflexivity</span>.<br/>
&nbsp;&nbsp;- <span class="id" title="tactic">apply</span> <a class="idref" href="Imp.html#E_IfFalse"><span class="id" title="constructor">E_IfFalse</span></a>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="Imp.html#E_Asgn"><span class="id" title="constructor">E_Asgn</span></a>. <span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
In the first step of the proof, we had to explicitly provide a
    longish expression to help Coq instantiate a "hidden" argument to
    the <span class="inlinecode"><span class="id" title="var">E_Seq</span></span> constructor.  This was needed because the definition
    of <span class="inlinecode"><span class="id" title="var">E_Seq</span></span>...
<br/>
<span class="inlinecode">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">E_Seq</span> : <span class="id" title="keyword">∀</span> <span class="id" title="var">c<sub>1</sub></span> <span class="id" title="var">c<sub>2</sub></span> <span class="id" title="var">st</span> <span class="id" title="var">st'</span> <span class="id" title="var">st''</span>,<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">st</span>  =[ <span class="id" title="var">c<sub>1</sub></span> ]=&gt; <span class="id" title="var">st'</span>  →<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">st'</span> =[ <span class="id" title="var">c<sub>2</sub></span> ]=&gt; <span class="id" title="var">st''</span> →<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">st</span>  =[ <span class="id" title="var">c<sub>1</sub></span> ; <span class="id" title="var">c<sub>2</sub></span> ]=&gt; <span class="id" title="var">st''</span>
</span>   is quantified over a variable, <span class="inlinecode"><span class="id" title="var">st'</span></span>, that does not appear in its
   conclusion, so unifying its conclusion with the goal state doesn't
   help Coq find a suitable value for this variable.  If we leave
   out the <span class="inlinecode"><span class="id" title="keyword">with</span></span>, this step fails ("Error: Unable to find an
   instance for the variable <span class="inlinecode"><span class="id" title="var">st'</span></span>").

<div class="paragraph"> </div>

   What's silly about this error is that the appropriate value for <span class="inlinecode"><span class="id" title="var">st'</span></span>
   will actually become obvious in the very next step, where we apply
   <span class="inlinecode"><span class="id" title="var">E_Asgn</span></span>.  If Coq could just wait until we get to this step, there
   would be no need to give the value explicitly.  This is exactly what
   the <span class="inlinecode"><span class="id" title="tactic">eapply</span></span> tactic gives us: 
</div>
<div class="code">

<span class="id" title="keyword">Example</span> <a id="ceval'_example1" class="idref" href="#ceval'_example1"><span class="id" title="definition">ceval'_example1</span></a>:<br/>
&nbsp;&nbsp;<a class="idref" href="Imp.html#empty_st"><span class="id" title="definition">empty_st</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">=[</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#X"><span class="id" title="definition">X</span></a> <a class="idref" href="Imp.html#91e9aa710642047a93142bdf557f1a1b"><span class="id" title="notation">:=</span></a> 2<a class="idref" href="Imp.html#313afe74ec81f2da17d8e7bca3b042e<sub>7</sub>"><span class="id" title="notation">;</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#:com:com_scope:'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">if</span></a> <a class="idref" href="Imp.html#5a96623d056c293ebbcdf03005796e<sub>09</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="Imp.html#X"><span class="id" title="definition">X</span></a> <a class="idref" href="Imp.html#d43f15edd52d29ae8e0f90730a25ed5c"><span class="id" title="notation">≤</span></a> 1<a class="idref" href="Imp.html#5a96623d056c293ebbcdf03005796e<sub>09</sub>"><span class="id" title="notation">)</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#:com:com_scope:'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">then</span></a> <a class="idref" href="Imp.html#Y"><span class="id" title="definition">Y</span></a> <a class="idref" href="Imp.html#91e9aa710642047a93142bdf557f1a1b"><span class="id" title="notation">:=</span></a> 3<br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#:com:com_scope:'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">else</span></a> <a class="idref" href="Imp.html#Z"><span class="id" title="definition">Z</span></a> <a class="idref" href="Imp.html#91e9aa710642047a93142bdf557f1a1b"><span class="id" title="notation">:=</span></a> 4<br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#:com:com_scope:'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">end</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="Imp.html#Z"><span class="id" title="definition">Z</span></a> <a class="idref" href="Maps.html#630986d105c0f1782d085b9a306379a<sub>7</sub>"><span class="id" title="notation">!<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>&gt;</span></span></span></a> 4 <a class="idref" href="Maps.html#630986d105c0f1782d085b9a306379a<sub>7</sub>"><span class="id" title="notation">;</span></a> <a class="idref" href="Imp.html#X"><span class="id" title="definition">X</span></a> <a class="idref" href="Imp.html#c78e6ed7f05f8146dc2bcb368255bece"><span class="id" title="notation">!<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>&gt;</span></span></span></a> 2<a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">)</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <a class="idref" href="Imp.html#E_Seq"><span class="id" title="constructor">E_Seq</span></a>. <span class="comment">(*&nbsp;1&nbsp;*)</span><br/>
&nbsp;&nbsp;- <span class="id" title="tactic">apply</span> <a class="idref" href="Imp.html#E_Asgn"><span class="id" title="constructor">E_Asgn</span></a>. <span class="comment">(*&nbsp;2&nbsp;*)</span><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="tactic">reflexivity</span>. <span class="comment">(*&nbsp;3&nbsp;*)</span><br/>
&nbsp;&nbsp;- <span class="comment">(*&nbsp;4&nbsp;*)</span> <span class="id" title="tactic">apply</span> <a class="idref" href="Imp.html#E_IfFalse"><span class="id" title="constructor">E_IfFalse</span></a>. <span class="id" title="tactic">reflexivity</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="Imp.html#E_Asgn"><span class="id" title="constructor">E_Asgn</span></a>. <span class="id" title="tactic">reflexivity</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
The <span class="inlinecode"><span class="id" title="tactic">eapply</span></span> <span class="inlinecode"><span class="id" title="var">H</span></span> tactic behaves just like <span class="inlinecode"><span class="id" title="tactic">apply</span></span> <span class="inlinecode"><span class="id" title="var">H</span></span> except
    that, after it finishes unifying the goal state with the
    conclusion of <span class="inlinecode"><span class="id" title="var">H</span></span>, it does not bother to check whether all the
    variables that were introduced in the process have been given
    concrete values during unification.

<div class="paragraph"> </div>

    If you step through the proof above, you'll see that the goal
    state at position <span class="inlinecode">1</span> mentions the <i>existential variable</i> <span class="inlinecode">?<span class="id" title="var">st'</span></span>
    in both of the generated subgoals.  The next step (which gets us
    to position <span class="inlinecode">2</span>) replaces <span class="inlinecode">?<span class="id" title="var">st'</span></span> with a concrete value.  This new
    value contains a new existential variable <span class="inlinecode">?<span class="id" title="var">n</span></span>, which is
    instantiated in its turn by the following <span class="inlinecode"><span class="id" title="tactic">reflexivity</span></span> step,
    position <span class="inlinecode">3</span>.  When we start working on the second
    subgoal (position <span class="inlinecode">4</span>), we observe that the occurrence of <span class="inlinecode">?<span class="id" title="var">st'</span></span>
    in this subgoal has been replaced by the value that it was given
    during the first subgoal. 
<div class="paragraph"> </div>

 Several of the tactics that we've seen so far, including <span class="inlinecode"><span class="id" title="tactic">∃</span></span>,
    <span class="inlinecode"><span class="id" title="tactic">constructor</span></span>, and <span class="inlinecode"><span class="id" title="tactic">auto</span></span>, have similar variants. The <span class="inlinecode"><span class="id" title="tactic">eauto</span></span>
    tactic works like <span class="inlinecode"><span class="id" title="tactic">auto</span></span>, except that it uses <span class="inlinecode"><span class="id" title="tactic">eapply</span></span> instead of
    <span class="inlinecode"><span class="id" title="tactic">apply</span></span>.  Tactic <span class="inlinecode"><span class="id" title="tactic">info_eauto</span></span> shows us which tactics <span class="inlinecode"><span class="id" title="tactic">eauto</span></span> uses
    in its proof search.

<div class="paragraph"> </div>

    Below is an example of <span class="inlinecode"><span class="id" title="tactic">eauto</span></span>.  Before using it, we need to give
    some hints to <span class="inlinecode"><span class="id" title="tactic">auto</span></span> about using the constructors of <span class="inlinecode"><span class="id" title="var">ceval</span></span>
    and the definitions of <span class="inlinecode"><span class="id" title="var">state</span></span> and <span class="inlinecode"><span class="id" title="var">total_map</span></span> as part of its
    proof search.

</div>
<div class="code">

<span class="id" title="keyword">Hint Constructors</span> <a class="idref" href="Auto.html#ceval"><span class="id" title="inductive">ceval</span></a> : <span class="id" title="var">core</span>.<br/>
<span class="id" title="keyword">Hint Transparent</span> <a class="idref" href="Imp.html#state"><span class="id" title="definition">state</span></a> <a class="idref" href="Maps.html#total_map"><span class="id" title="definition">total_map</span></a> : <span class="id" title="var">core</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Example</span> <a id="eauto_example" class="idref" href="#eauto_example"><span class="id" title="definition">eauto_example</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="s':126" class="idref" href="#s':126"><span class="id" title="binder">s'</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="Imp.html#Y"><span class="id" title="definition">Y</span></a> <a class="idref" href="Maps.html#630986d105c0f1782d085b9a306379a<sub>7</sub>"><span class="id" title="notation">!<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>&gt;</span></span></span></a> 1 <a class="idref" href="Maps.html#630986d105c0f1782d085b9a306379a<sub>7</sub>"><span class="id" title="notation">;</span></a> <a class="idref" href="Imp.html#X"><span class="id" title="definition">X</span></a> <a class="idref" href="Imp.html#c78e6ed7f05f8146dc2bcb368255bece"><span class="id" title="notation">!<span class="nowrap"><span style='font-size:85%;'><span style='vertical-align:5%;'><span style='letter-spacing:-.2em;'>-</span></span>&gt;</span></span></span></a> 2<a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">)</span></a> <a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">=[</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#:com:com_scope:'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">if</span></a> <a class="idref" href="Imp.html#5a96623d056c293ebbcdf03005796e<sub>09</sub>"><span class="id" title="notation">(</span></a><a class="idref" href="Imp.html#X"><span class="id" title="definition">X</span></a> <a class="idref" href="Imp.html#d43f15edd52d29ae8e0f90730a25ed5c"><span class="id" title="notation">≤</span></a> <a class="idref" href="Imp.html#Y"><span class="id" title="definition">Y</span></a><a class="idref" href="Imp.html#5a96623d056c293ebbcdf03005796e<sub>09</sub>"><span class="id" title="notation">)</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#:com:com_scope:'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">then</span></a> <a class="idref" href="Imp.html#Z"><span class="id" title="definition">Z</span></a> <a class="idref" href="Imp.html#91e9aa710642047a93142bdf557f1a1b"><span class="id" title="notation">:=</span></a> <a class="idref" href="Imp.html#Y"><span class="id" title="definition">Y</span></a> <a class="idref" href="Imp.html#:com::x_'-'_x"><span class="id" title="notation">-</span></a> <a class="idref" href="Imp.html#X"><span class="id" title="definition">X</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#:com:com_scope:'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">else</span></a> <a class="idref" href="Imp.html#Y"><span class="id" title="definition">Y</span></a> <a class="idref" href="Imp.html#91e9aa710642047a93142bdf557f1a1b"><span class="id" title="notation">:=</span></a> <a class="idref" href="Imp.html#X"><span class="id" title="definition">X</span></a> <a class="idref" href="Imp.html#ad27fd091d9c8494ce7e5135135d19a<sub>8</sub>"><span class="id" title="notation">+</span></a> <a class="idref" href="Imp.html#Z"><span class="id" title="definition">Z</span></a><br/>
&nbsp;&nbsp;&nbsp;&nbsp;<a class="idref" href="Imp.html#:com:com_scope:'if'_x_'then'_x_'else'_x_'end'"><span class="id" title="notation">end</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Imp.html#6037d1f4787d63984beccd5510872ee<sub>9</sub>"><span class="id" title="notation">]=&gt;</span></a> <a class="idref" href="Auto.html#s':126"><span class="id" title="variable">s'</span></a>.<br/>
<span class="id" title="keyword">Proof</span>. <span class="id" title="tactic">info_eauto</span>. <span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
The <span class="inlinecode"><span class="id" title="tactic">eauto</span></span> tactic works just like <span class="inlinecode"><span class="id" title="tactic">auto</span></span>, except that it uses
    <span class="inlinecode"><span class="id" title="tactic">eapply</span></span> instead of <span class="inlinecode"><span class="id" title="tactic">apply</span></span>; <span class="inlinecode"><span class="id" title="tactic">info_eauto</span></span> shows us which facts
    <span class="inlinecode"><span class="id" title="tactic">eauto</span></span> uses. 
<div class="paragraph"> </div>

 Pro tip: One might think that, since <span class="inlinecode"><span class="id" title="tactic">eapply</span></span> and <span class="inlinecode"><span class="id" title="tactic">eauto</span></span>
    are more powerful than <span class="inlinecode"><span class="id" title="tactic">apply</span></span> and <span class="inlinecode"><span class="id" title="tactic">auto</span></span>, we should just use them
    all the time.  Unfortunately, they are also significantly slower
 especially <span class="inlinecode"><span class="id" title="tactic">eauto</span></span>.  Coq experts tend to use <span class="inlinecode"><span class="id" title="tactic">apply</span></span> and <span class="inlinecode"><span class="id" title="tactic">auto</span></span>
    most of the time, only switching to the <span class="inlinecode"><span class="id" title="var">e</span></span> variants when the
    ordinary variants don't do the job. 
<div class="paragraph"> </div>

<a id="lab407"></a><h1 class="section">Constraints on Existential Variables</h1>

<div class="paragraph"> </div>

 In order for <span class="inlinecode"><span class="id" title="keyword">Qed</span></span> to succeed, all existential variables need to
    be determined by the end of the proof. Otherwise Coq
    will (rightly) refuse to accept the proof. Remember that the Coq
    tactics build proof objects, and proof objects containing
    existential variables are not complete. 
</div>
<div class="code">

<span class="id" title="keyword">Lemma</span> <a id="silly1" class="idref" href="#silly1"><span class="id" title="lemma">silly1</span></a> : <span class="id" title="keyword">∀</span> (<a id="P:127" class="idref" href="#P:127"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<a id="Q:128" class="idref" href="#Q:128"><span class="id" title="binder">Q</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>),<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:129" class="idref" href="#x:129"><span class="id" title="binder">x</span></a> <a id="y:130" class="idref" href="#y:130"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="Auto.html#P:127"><span class="id" title="variable">P</span></a> <a class="idref" href="Auto.html#x:129"><span class="id" title="variable">x</span></a> <a class="idref" href="Auto.html#y:130"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:131" class="idref" href="#x:131"><span class="id" title="binder">x</span></a> <a id="y:132" class="idref" href="#y:132"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="Auto.html#P:127"><span class="id" title="variable">P</span></a> <a class="idref" href="Auto.html#x:131"><span class="id" title="variable">x</span></a> <a class="idref" href="Auto.html#y:132"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#Q:128"><span class="id" title="variable">Q</span></a> <a class="idref" href="Auto.html#x:131"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#Q:128"><span class="id" title="variable">Q</span></a> 42.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">P</span> <span class="id" title="var">Q</span> <span class="id" title="var">HP</span> <span class="id" title="var">HQ</span>. <span class="id" title="tactic">eapply</span> <span class="id" title="var">HQ</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">HP</span>.<br/>
</div>

<div class="doc">
Coq gives a warning after <span class="inlinecode"><span class="id" title="tactic">apply</span></span> <span class="inlinecode"><span class="id" title="var">HP</span></span>: "All the remaining goals
    are on the shelf," means that we've finished all our top-level
    proof obligations but along the way we've put some aside to be
    done later, and we have not finished those.  Trying to close the
    proof with <span class="inlinecode"><span class="id" title="keyword">Qed</span></span> would yield an error. (Try it!) 
</div>
<div class="code">
<span class="id" title="keyword">Abort</span>.<br/>
</div>

<div class="doc">
An additional constraint is that existential variables cannot be
    instantiated with terms containing ordinary variables that did not
    exist at the time the existential variable was created.  (The
    reason for this technical restriction is that allowing such
    instantiation would lead to inconsistency of Coq's logic.) 
</div>
<div class="code">

<span class="id" title="keyword">Lemma</span> <a id="silly2" class="idref" href="#silly2"><span class="id" title="lemma">silly2</span></a> :<br/>
&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="P:133" class="idref" href="#P:133"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<a id="Q:134" class="idref" href="#Q:134"><span class="id" title="binder">Q</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>),<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="y:135" class="idref" href="#y:135"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="Auto.html#P:133"><span class="id" title="variable">P</span></a> 42 <a class="idref" href="Auto.html#y:135"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:136" class="idref" href="#x:136"><span class="id" title="binder">x</span></a> <a id="y:137" class="idref" href="#y:137"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="Auto.html#P:133"><span class="id" title="variable">P</span></a> <a class="idref" href="Auto.html#x:136"><span class="id" title="variable">x</span></a> <a class="idref" href="Auto.html#y:137"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#Q:134"><span class="id" title="variable">Q</span></a> <a class="idref" href="Auto.html#x:136"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#Q:134"><span class="id" title="variable">Q</span></a> 42.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">P</span> <span class="id" title="var">Q</span> <span class="id" title="var">HP</span> <span class="id" title="var">HQ</span>. <span class="id" title="tactic">eapply</span> <span class="id" title="var">HQ</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">HP</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">y</span> <span class="id" title="var">HP'</span>].<br/>
&nbsp;&nbsp;<span class="id" title="var">Fail</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">HP'</span>.<br/>
</div>

<div class="doc">
The error we get, with some details elided, is:
<br/>
<span class="inlinecode">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<span class="id" title="var">cannot</span> <span class="id" title="tactic">instantiate</span> "?y" <span class="id" title="var">because</span> "y" <span class="id" title="keyword">is</span> <span class="id" title="var">not</span> <span class="id" title="keyword">in</span> <span class="id" title="var">its</span> <span class="id" title="var">scope</span>
</span>    In this case there is an easy fix: doing <span class="inlinecode"><span class="id" title="tactic">destruct</span></span> <span class="inlinecode"><span class="id" title="var">HP</span></span> <i>before</i>
    doing <span class="inlinecode"><span class="id" title="tactic">eapply</span></span> <span class="inlinecode"><span class="id" title="var">HQ</span></span>. 
</div>
<div class="code">
<span class="id" title="keyword">Abort</span>.<br/><hr class='doublespaceincode'/>
<span class="id" title="keyword">Lemma</span> <a id="silly2_fixed" class="idref" href="#silly2_fixed"><span class="id" title="lemma">silly2_fixed</span></a> :<br/>
&nbsp;&nbsp;<span class="id" title="keyword">∀</span> (<a id="P:138" class="idref" href="#P:138"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<a id="Q:139" class="idref" href="#Q:139"><span class="id" title="binder">Q</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>),<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="y:140" class="idref" href="#y:140"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="Auto.html#P:138"><span class="id" title="variable">P</span></a> 42 <a class="idref" href="Auto.html#y:140"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:141" class="idref" href="#x:141"><span class="id" title="binder">x</span></a> <a id="y:142" class="idref" href="#y:142"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="Auto.html#P:138"><span class="id" title="variable">P</span></a> <a class="idref" href="Auto.html#x:141"><span class="id" title="variable">x</span></a> <a class="idref" href="Auto.html#y:142"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#Q:139"><span class="id" title="variable">Q</span></a> <a class="idref" href="Auto.html#x:141"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#Q:139"><span class="id" title="variable">Q</span></a> 42.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">P</span> <span class="id" title="var">Q</span> <span class="id" title="var">HP</span> <span class="id" title="var">HQ</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">HP</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">y</span> <span class="id" title="var">HP'</span>].<br/>
&nbsp;&nbsp;<span class="id" title="tactic">eapply</span> <span class="id" title="var">HQ</span>. <span class="id" title="tactic">apply</span> <span class="id" title="var">HP'</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
The <span class="inlinecode"><span class="id" title="tactic">apply</span></span> <span class="inlinecode"><span class="id" title="var">HP'</span></span> in the last step unifies the existential variable
    in the goal with the variable <span class="inlinecode"><span class="id" title="var">y</span></span>.

<div class="paragraph"> </div>

    Note that the <span class="inlinecode"><span class="id" title="tactic">assumption</span></span> tactic doesn't work in this case, since
    it cannot handle existential variables.  However, Coq also
    provides an <span class="inlinecode"><span class="id" title="var">eassumption</span></span> tactic that solves the goal if one of
    the premises matches the goal up to instantiations of existential
    variables. We can use it instead of <span class="inlinecode"><span class="id" title="tactic">apply</span></span> <span class="inlinecode"><span class="id" title="var">HP'</span></span> if we like. 
</div>
<div class="code">

<span class="id" title="keyword">Lemma</span> <a id="silly2_eassumption" class="idref" href="#silly2_eassumption"><span class="id" title="lemma">silly2_eassumption</span></a> : <span class="id" title="keyword">∀</span> (<a id="P:143" class="idref" href="#P:143"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<a id="Q:144" class="idref" href="#Q:144"><span class="id" title="binder">Q</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>),<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="y:145" class="idref" href="#y:145"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="Auto.html#P:143"><span class="id" title="variable">P</span></a> 42 <a class="idref" href="Auto.html#y:145"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:146" class="idref" href="#x:146"><span class="id" title="binder">x</span></a> <a id="y:147" class="idref" href="#y:147"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="Auto.html#P:143"><span class="id" title="variable">P</span></a> <a class="idref" href="Auto.html#x:146"><span class="id" title="variable">x</span></a> <a class="idref" href="Auto.html#y:147"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#Q:144"><span class="id" title="variable">Q</span></a> <a class="idref" href="Auto.html#x:146"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#Q:144"><span class="id" title="variable">Q</span></a> 42.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">P</span> <span class="id" title="var">Q</span> <span class="id" title="var">HP</span> <span class="id" title="var">HQ</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">HP</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">y</span> <span class="id" title="var">HP'</span>]. <span class="id" title="tactic">eapply</span> <span class="id" title="var">HQ</span>. <span class="id" title="var">eassumption</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
</div>

<div class="doc">
The <span class="inlinecode"><span class="id" title="tactic">eauto</span></span> tactic will use <span class="inlinecode"><span class="id" title="tactic">eapply</span></span> and <span class="inlinecode"><span class="id" title="var">eassumption</span></span>, streamlining
    the proof even further. 
</div>
<div class="code">

<span class="id" title="keyword">Lemma</span> <a id="silly2_eauto" class="idref" href="#silly2_eauto"><span class="id" title="lemma">silly2_eauto</span></a> : <span class="id" title="keyword">∀</span> (<a id="P:148" class="idref" href="#P:148"><span class="id" title="binder">P</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>) (<a id="Q:149" class="idref" href="#Q:149"><span class="id" title="binder">Q</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <span class="id" title="keyword">Prop</span>),<br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">∃</span></a> <a id="y:150" class="idref" href="#y:150"><span class="id" title="binder">y</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#a883bdd010993579f99d60b3775bcf54"><span class="id" title="notation">,</span></a> <a class="idref" href="Auto.html#P:148"><span class="id" title="variable">P</span></a> 42 <a class="idref" href="Auto.html#y:150"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <a id="x:151" class="idref" href="#x:151"><span class="id" title="binder">x</span></a> <a id="y:152" class="idref" href="#y:152"><span class="id" title="binder">y</span></a> : <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Datatypes.html#nat"><span class="id" title="inductive">nat</span></a>, <a class="idref" href="Auto.html#P:148"><span class="id" title="variable">P</span></a> <a class="idref" href="Auto.html#x:151"><span class="id" title="variable">x</span></a> <a class="idref" href="Auto.html#y:152"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a> <a class="idref" href="Auto.html#Q:149"><span class="id" title="variable">Q</span></a> <a class="idref" href="Auto.html#x:151"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/library//Coq.Init.Logic.html#::type_scope:x_'-&gt;'_x"><span class="id" title="notation">→</span></a><br/>
&nbsp;&nbsp;<a class="idref" href="Auto.html#Q:149"><span class="id" title="variable">Q</span></a> 42.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
&nbsp;&nbsp;<span class="id" title="tactic">intros</span> <span class="id" title="var">P</span> <span class="id" title="var">Q</span> <span class="id" title="var">HP</span> <span class="id" title="var">HQ</span>. <span class="id" title="tactic">destruct</span> <span class="id" title="var">HP</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">y</span> <span class="id" title="var">HP'</span>]. <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/><hr class='doublespaceincode'/>
<span class="comment">(*&nbsp;2021-08-11&nbsp;15:08&nbsp;*)</span><br/>
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